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Compute:
(i)
(ii)
(iii)
Prove that LCM {6!, 7!, 8!} = 8!
Prove that .
If , find the value of x.
Write the following products in factorial notation:
(i) 6 × 7 × 8 × 9 × 10 × 11 × 12
(ii) 3 × 6 × 9 × 12 × 15
Which of the following are true of false?
(i) (2 + 3) ! = 2 ! + 3!
(ii) (2 × 3)! = (2!) × (3!)
If (n + 1) ! = 12 × (n – 1) !, find the value of n.
If (n + 2) ! = 2550 × n!, find the value of n.
If (n + 3) ! = 56 × (n + 1) !, find the value of n.
If = 2 : 1, find the value of n.
If = 44 : 3, find the value of n.
Evaluate , when n = 15 and r = 12.
Prove that (n + 2) × (n!) + (n + 1) ! = (n!). (2n + 3)
Prove that
(i) = n(n – 1) (n – 2) …. (r + 1)
(ii) (n – r + 1).
There are 10 buses running between Delhi and Agra. In how many ways can a man go from Delhi to Agra and return by a different bus?
A, B and C are three cities. There are 5 routes from A to B and 3 routes from B to C. How many different routes are there from A to C via B?
There are 12 steamers plying between A and B. In how many ways could the round trip from A be made if the return was made on (i) the same steamer? (ii) a different steamer?
In How many ways can 4 people be seated in a row containing 5 seats?
In How many ways can 5 ladies draw water from 5 taps, assuming the no tap remains unused?
In a textbook on mathematics there are three exercises A, B and C consisting of 12, 18 and 10 questions respectively. In how many ways can three questions be selected choosing one from each exercise?
In a school, there are four sections of 40 students each in XI standard. In how many ways can a set of 4 student representatives be chosen, one from each section?
In how many ways can a vowel, a consonant and a digit be chosen out of the 26 letters of the English alphabet and the 10 digits?
How many 8-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 270 and n digit appears more than once?
(ac and the outcomes are recorded. How many possible outcomes are there?
(b) How many possible outcomes if the coin is tossed.
(i) four times? (ii) five times? (iii) n times?
Find the number of different signals that can be generated by arranging at least 2 flags in order (one below the other) on a vertical staff, if five different flags are available.
How many 4-letter codes can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?
Given, A = {2, 3, 5} and B = {0, 1}. Find the number of different ordered pairs in which the first entry is an element of A and the second is an element of B.
How many arithmetic progressions with 10 terms are there whose first term in the set {1, 2, 3} and whose common difference is in the set {2, 3, 4}?
There are 6 items in column A and 6 items in column B. A student is asked to match each item in column A with an item in column B. How many possible (correct or incorrect) answers are there to this question?
A mint prepares metallic calendars specifying months, dates and days in the form of monthly sheets (one plate for each month). How many types of February calendars should it prepare to serve for all the possibilities in the future years?
From among the 36 teachers in a school, one principal and one vice-principal are to be appointed. In how many ways can this be done?
A sample of 3 bulbs is tested. A bulb is labeled as G if it is good and D if it is defective. Find the number of all possible outcomes.
For a set of five true or false questions, no student has written the all correct answer and no two students have given the same sequence of answers. What is the maximum number of students in the class for this to be possible?
In how many ways can the following prizes be given away to a class of 20 students :first and second in mathematics; first and second in chemistry; first in physics and first in English?
Find the total number of ways of answering 5 objective-type question, each question having 4 choices.
A gentleman has 6 friends to invite. In how many ways can be send invitation cards to them, if he has 3 servants to carry the cards?
In how many ways 6 rings of different types can be worn in 4 fingers?
In how many ways can 5 letters be posted in 4 letter boxes?
How many 3-letters words can be formed using a, b, c, d, e if
(i) repetition of letters is not allowed?
(ii) repetition of letters is allowed
How many 4-digit numbers are there, when a digit may be repeated any number of times?
How many numbers can be formed from the digits 1, 3, 5, 9 if repetition of digits is not allowed?
How many 3-digit numbers are there with no digit repeated?
How many 3-digit numbers can be formed by using the digits 0, 1, 3, 5, 7 while each digit may be repeated any number of times?
How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7, 9 when no digit is repeated? How many of them are divisible by 10?
How many natural numbers less than 1000 can be formed from the digits 0, 1, 2, 3, 4, 5 when a digit may be repeated any number of times?
How many 6-digit telephone numbers can be constructed using the digits 0 to 9, if each number starts with 67 and no digit appears more than once?
In how many ways can three jobs, I, II and III be assigned to three persons A, B and C if one person is assigned only one job and all are capable of doing each job?
A number lock on a suitcase has three wheels each labeled with ten digits 0 to 9. if opening of the lock is a particular sequence of three digits with no repeats, how many such sequences will be possible? Also, find the number of unsuccessful attempts to open the lock.
A customer forgets a four-digit code for an automated teller machine (ATM) in a bank. However, he remembers that this code consists of digits 3, 5, 6, 9. Find the largest possible number of trials necessary to obtain the correct code.
In how many ways can 3 prizes be distributed among 4 girls, when
(i) no girl gets more than one prize?
(ii) a girl may get any number of prizes?
(iii) no girl gets all the prizes?
Evaluate:
10P4
62P3
6P6
9P0
Prove that 9P3 + 3 × 9P2 = 10P3.
(i) If nP5 = 20 × nP3, find n.
(ii) If 16 × nP3 = 13 × n+1P3, find n.
(iii) If 2nP3 = 100 × nP2, find n.
(i) If If 5Pr = 2 × 6Pr–1, find r.
(ii) If 20Pr = 13 × 20Pr–1, find r.
(iii) If 11Pr = 12Pr–1, find r.
(i) If nP4 : nP5, = 1 : 2, find n.
(ii) If n–1P3 : n+1P3, = 5 : 12, find n.
If 15Pr–1 : 16Pr–2, = 3 : 4, find r.
If 2n–1Pn : 2n+1Pn–1, = 22 : 7, find n.
Find n, If n+5Pn+1 = (n – 1). n+3Pn, find n.
Prove that 1 + 1. 1P1 + 2. 2P2 + 3. 3P3 + …. n. nPn = n+1Pn+1.
Find the number of permutations of 10 objects, taken 4 at a time.
In how many ways can 5 persons occupy 3 vacant seats?
In how many ways can 7 people line up at a ticket window of a cinema hall?
In how many ways can 5 children stand in a queue?
In how many ways can 6 women draw water from 6 wells if no well remains unused?
In how many ways can 4 different books, one each in chemistry, physics, biology and mathematics, be arranged on a shelf?
Six students are contesting the election for the presidentship of the students, union. In how many ways can their names be listed on the ballot papers?
It is required to seat 5 men and 3 women in a row so that the women occupy the even places. How many such arrangements are possible?
There are 6 items in column A and 6 items in column B. A student is asked to match each item in column A with an item in column B. How many possible, correct or incorrect answers are there to this question?
Five letters F, K, R, R and V one in each were purchased from a plastic warehouse. How many ordered pairs of letters, to be used as initials, can be formed from them?
Ten students are participating in a race. In how many ways can the first three prizes be won?
If there are 6 periods on each working day of a school, in how many ways can one arrange 5 subjects such that each subject is allowed at least one period?
In how many ways can 6 pictures be hung from 4 picture nails on a wall?
Find the number of words formed (may be meaningless) by using all the letters of the word ‘EQUATION’, using each letter exactly once.
Find the number of different 4-letter words (may be meaningless) that can be formed from the letters of the word ‘NUMBERS’,
How many words can be formed from the letters of the word ‘SUNDAY’? How many of these begin with D?
How many words beginning with C and ending with Y can be formed by using the letters of the word ‘COURTESY’?
Find the number of permutations of the letters of the word ‘ENGLISH’. How many of these begin with E and end with I?
In how many ways can the letters of the word ‘HEXAGON’ be permuted? In how many words will the vowels be together?
How many words can be formed out of the letters of the word ‘ORIENTAL’ so that the vowels always occupy the odd places?
In how many ways can the letters of the word ‘FAILURE’ be arranged so that the consonants may occupy only odd positions?
In how many arrangements of the word ‘GOLDEN’ will the vowels never occur together?
Find the number of ways in which the letters of the word ‘MACHINE’ can be arranged such that the vowels may occupy only odd positions.
How many permutations can be formed by the letters of the word ‘VOWELS’, when
(i) there is no restriction on letters;
(ii) each word begins with E;
(iii) each word begins with O and ends with L;
(iv) all vowels come together;
(v) all consonants come together?
How many numbers divisible by 5 and lying between 3000 and 4000 can be formed by using the digits 3, 4, 5, 6, 7, 8 when no digit is repeated in any such number?
In an examination, there are 8 candidates out of which 3 candidates have to appear in mathematics and the rest in different subjects. In how many ways can they are seated in a row if candidates appearing in mathematics are not to sit together?
In how many ways can 5 children be arranged in a line such that
(i) two of them, Rajan and Tanvy, are always together?
(ii) two of them, Rajan and Tanvy, are never together,
when a group photograph is taken, all the seven teachers should be in the first row, and all the twenty students should be in the second row. If the tow corners of the second row are reserved for the two tallest students, interchangeable only between them, and if the middle seat of the front row is reserved for the principal, how many arrangements are possible?
Find the number of ways in which m boys and n girls may be arranged in a row so that no two of the girls are together; it is given that m > n.
Find the total number of permutations of the letters of each of the words given below:
(i) APPLE (ii) ARRANGE
(iii) COMMERCE (iv) INSTITUTE
(v) ENGINEERING
(vi) INTERMEDIATE
In how many ways can the letters of the expression x2y2z4 be arranged when written without using exponents?
There are 3 blue balls, 4 red balls and 5 green balls. In how many ways can they are arranged in a row?
A child has three plastic toys bearing the digits 3, 3, 5 respectively. How many 3 - digit numbers can he make using them?
How many different signals can be transmitted by arranging 2 red, 3 yellow and 2 green flags on a pole, if all the seven flags are used to transmit a signal?
How many words can be formed by arranging the letters of the word ‘ARRANGEMENT’, so that the vowels remain together?
How many words can be formed by arranging the letters of the word ‘INDIA’, so that the vowels are never together?
Find the number of arrangements of the letters of the word ‘ALGEBRA’ without altering the relative positions of the vowels and the consonants.
How many words can be formed from the letters of the word ‘SERIES’, which start with S and end with S?
In how many ways can the letters of the word ‘PARALLEL’ be arranged so that all L’s do not come together?
How many different words can be formed with the letters of the word ‘CAPTAIN’? In how many of these C and T are never together?
In how many ways can the letters of the word ‘ASSASSINATION’ be arranged so that all S’s are together?
(i)How many arrangements can be made by using all the letters of the word ‘MATHEMATICS’?
(ii) How many of them begin with C?
(iii) How many of them begin with T?
In how many ways can the letters of the word ‘INTERMEDIATE’ be arranged so that:
(i) the vowels always occupy even places?
(ii) the relative orders of vowels and consonants do not change?
(i)Find the number of different words by using all the letters of the word, ‘INSTITUTION’.
In how many of them
(ii) are the three T’ s together
(iii) are the first two letters the two N’ s?
How many five - digit numbers can be formed with the digits 5, 4, 3, 5, 3?
How many numbers can be formed with the digits 2, 3, 4, 5, 4, 3, 2 so that the odd digits occupy the odd places?
How many 7 - digit numbers can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?
How many 6 - digit numbers can be formed by using the digits 4, 5, 0, 3, 4, 5?
The letters of the word ‘INDIA’ are arranged as in a dictionary. What are the 1st, 13th, 49th and 60th words?
A child has 6 pockets. In how many ways, he can put 5 marbles in his pocket?
In how many ways can 5 bananas be distributed among 3 boys, there being no restriction to the number of bananas each boy
may get?
In how many ways can 3 letters can be posted in 2 letterboxes?
How many 3-digit numbers are there when a digit may be repeated any numbers of time?
How many 4-digit numbers can be formed with the digits 0, 2, 3, 4, 5 when a digit may be repeated any numbers of time in any arrangement?
In how many ways can 4 prizes be given to 3 boys when a boy is eligible for all prizes?
There are 4 candidates for the post of a chairman, and one is to be elected by votes of 5 men. In how many ways can the vote be given?
In how many ways can 6 persons be arranged in
(i) a line, (ii) a circle?
There are 5 men and 5 ladies to dine at a round table. In how many ways can they sit so that no ladies are together?
In how many ways can 11 members of a committee sit at a round table so that the secretary and the joint secretary are always the neighbours of the president?
In how many ways can 8 persons be seated at a round table so that all shall not have the same neighbours in any two arrangement?
In how many differents ways can 20 different pearls be arranged to form a necklace?
In how many different ways can a garland of 16 different flowers be made?
If (n + 1)! = 12 × [(n – 1)!], find the value of n.
If, find the value of x.
How many 3-digit numbers above 600 can be formed by using the digits 2, 3, 4, 5, 6, if repetition of digits is allowed?
How many numbers divisible by 5 and lying between 4000 and 5000 can be formed from the digits 4, 5, 6, 7, 8 if repetition of digits is allowed?
In how many ways can the letters of the word ‘CHEESE’ be arranged?
In how many ways can the letters of the word ‘PERMUTATIONS’ be arranged if each word starts with P and ends with S?
How many different words can be formed by using all the letters of the word ‘ALLAHABAD’?
How many permutations of the letters of the word ‘APPLE’ are there?
How many words can be formed by the letters of the word ‘SUNDAY’?
In how many ways can 4 letters be posted in 5 letter boxes?
In how many ways can 4 women draw water from 4 taps if no tap remains unused?
How many 5-digit numbers can be formed by using the digits 0, 1 and 2?
In how many ways can 5 boys and 3 girls be seated in a row so that each girl is between 2 boys?
A child has plastic toys bearing the digits 4, 4 and 5. How many 3-digit numbers can he make using them?
In how many ways can the letters of the word ‘PENCIL’ be arranged so that N is always next to E?
