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Evaluate:

^{20}C_{4}

^{16}C_{13}

^{90}C_{88}

^{71}C_{71}

^{n+1}C_{n}

Verify that:

(i) ^{15}C_{8} + ^{15}C_{9} – ^{15}C_{6} – ^{15}C_{7} = 0

(ii) ^{10}C_{4} + ^{10}C_{3} = ^{11}C_{4}

(i) If ^{n}C_{7} = ^{n}C_{5}, find n.

(ii) If ^{n}C_{14} = ^{n}C_{16}, find ^{n}C_{28}.

(iii) If ^{n}C_{16} = ^{n}C_{14}, find ^{n}C_{27}.

(i) If ^{20}C_{r} = ^{20}C_{r+6}, find r.

(ii) If ^{18}C_{r} = ^{18}C_{r+2}, find ^{r}C_{5}.

If ^{n}C_{r–1} = ^{n}C_{3r}, find r.

If ^{2n}C_{3}: ^{n}C_{3} = 12 : 1, find n.

If ^{15}C_{r} : ^{15}C_{r–1} = 11 : 5, find r.

If ^{n}P_{r} = 840 and ^{n}C_{r} = 35, find the value of r.

If ^{n}C_{r–1} = 36, ^{n}C_{r} = 84 and ^{n}C_{r+1} = 126, find r.

If ^{n+1}C_{r+1} : ^{n}C_{r} = 11 : 6 and ^{n}C_{r} : ^{n–1}C_{r–1} = 6 : 3, find n and r.

How many different teams of 11 players can be chosen from 15 players?

If there are 12 persons in a party and if each two of them shake hands with each other, how many handshakes are possible?

How many chords can be drawn through 21 points on a circle?

From a class of 25 students, 4 are to be chosen for a competition. In how many ways can this be done?

In how many ways can 5 sportsmen be selected from a group of 10?

A bag contains 5 black and 6 red balls. Find the number of ways in which 2 black and 3 red balls can be selected.

Find the number of ways of selecting 9 balls from 6 red balls, 5 while balls and 4 blue balls if each selection consists of 3 balls of each colour.

How many different boat parties of 8 consisting of 5 boys and 3 girls can be made from 20 boys and 10 girls.

In How many ways can a student chose 5 courses out of 9 courses if 2 specific courses are compulsory for every student?

A sports team of 11 students is to be constituted, choosing at least 5 from class XI and at least 5 from class XII. If there are 20 students in each of these classes, in how many ways can the team be constituted?

From 4 officers and 8 clerks, in how many ways can 6 be chosen (i) to include exactly one officer, (ii) to include at least one officer?

A cricket team of 11 players is to be selected from 16 players including 5 bowlers and 2 wicketkeepers. In how many ways can a team be selected so as to consist of exactly 3 bowlers and 1 wicketkeeper?

In how many ways can a cricket team be selected from a group of 25 players containing 10 batsmen, 8 bowlers, 5 all-rounders and 2 wicketkeepers, assuming that the team of 11 players requires 5 batsmen, 3 all-rounders, 2 bowlers and 1 wicketkeeper?

A question paper has two parts, part A and part B, each containing 10 questions. If the student has to choose 8 from part A and 5 from part B, in how many ways can he choose the questions?

In an examination, a student has to answer 4 questions out of 5. Questions 1 and 2 are compulsory. Find the number of ways in which the student can make a choice.

In an examination, a student has to answer 10 questions, choosing at least 4 from each of part A and part B. If there are 6 questions in part A and 7 in part B, in how many ways can these questions be chosen?

In an examination, a candidate is required to answer 7 questions out of 12, which are divided into two groups, each containing 6 questions. One cannot attempt more than 5 questions from either group. In how many ways can he choose these questions?

Out of 6 teachers and 8 students, a committee of 11 is being formed. In how many ways can this be done, if the committee contains

(i) exactly 4 teachers?

(ii) at least 4 teachers?

A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of

(i) exactly 3 girls?

(ii) at least 3 girls?

(iii) at most 3 girls?

A committee of three persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done? How many of these committees would consist of 1 man and 2 women?

A committee of 5 is to be formed out of 6 gents and 4 ladies. In how many ways can this be done, when

(i) at least 2 ladies are included?

(ii) at most 2 ladies are included?

From a class of 14 boys and 10 girls, 10 students are to be chosen for a competition, at least including 4 boys and 4 girls. The 2 girls who won the prizes last year should be included. In how many ways can the selection be made?

Find the number of 5-card combinations out of a deck of 52 cards if a least one of the five cards has to be king.

Find the number of diagonals of

(i) a hexagon,

(ii) a decagon,

(iii) a polygon of 18 sides

How many triangles can be obtained by joining 12 points, four of which are collinear?

How many triangles can be formed in a decagon?

How many different selections of 4 books can be made from 10 different books, if

(i) there is no restriction?

(ii) two particular books are always selected?

(iii) two particular books are never selected?

How many different products can be obtained by multiplying two or more of the numbers 3, 5, 7, 11 without repetition?

Find the number of ways in which a committee of 2 teachers and 3 students can be formed out of 10 teachers and 20 students. In how many of these committees

(i) a particular teacher is included?

(ii) a particular student is included?

(iii) a particular student is excluded?

There are 18 points in a plane of which 5 are collinear. How many straight lines can be formed by joining them?

Out of 12 consonants and 5 vowels, how many words, each containing 3 consonants and 2 vowels, can be formed?

How many words, each of 3 vowels and 2 consonants, can be formed from the letters of the word ‘INVOLUTE’?

The English alphabet has 21 consonants and 5 vowels. How many words with two different consonants and three different vowels can be formed from the alphabet?

In how many ways can 4 girls and 3 boys be seated in a row so that no two boys are together?

How many words, with or without meaning, can be formed from the letters of the word, ‘MONDAY’, assuming that no letter is repeated, if (i) 4 letters are used at a time? (ii) all letters are used at a time? (iii) all letters are used, but the first letter is a vowel?

If ^{20}C_{r} = ^{20}C_{r–10} then find the value of ^{17}C_{r}.

If ^{20}C_{r+1} = ^{20}C_{r–10} then find the value of ^{10}C_{r}.

If ^{n}C_{r+1} = ^{n}C_{8} then find the value of ^{22}C_{n}.

If ^{35}C_{n+7} = ^{35}C_{4n–2} then find the value of n.

Find the values of (i) ^{200}C_{198}, (ii) ^{76}C_{0}, (iii) ^{15}C_{15}.

If ^{m}C_{1} = ^{n}C_{2} prove that m = n(n – 1).

Write the value of (^{5}C_{1} + ^{5}C_{2} + ^{5}C_{3} + ^{5}C_{4} + ^{5}C_{5}).

If ^{n+1}C_{3} = 2(^{n}C_{2}), find the value of n.

If ^{n}P_{r} = 720 and ^{n}C_{r} = 120 then find the value of r.

If ^{(n2–n)}C_{2} = ^{(n2–n)}C_{4} = 120 then find the value of n.

How many words are formed by 2 vowels and 3 consonants, taken from 4 vowels and 5 consonants?

Find the number of diagonals in an n-sided polygon.

Three persons enter a railway compartment having 5 vacant seats. In how many ways can they seat themselves?

There are 12 points in a plane, out of which 3 points are collinear. How many straight lines can be drawn by joining any two of them?

In how many ways can committee of 5 be made out of 6 men and 4 women, containing at least 2 women?

There are 13 cricket players, out of which 4 are bowlers. In how many ways can team of 11 be selected from them so as to include at least 3 bowlers?

How many different committees of 5 can be formed from 6 men and 4 women, if each committee consists of 3 men and 2 women?

How many parallelograms can be formed from a set of 4 parallel lines interesting another set of 3 parallel lines?