Given:

- Total number of students = 100

- Number of students studying English(E) only = 18

- Number of students learning English but not Hindi(H) = 23

- Number of students learning English and Sanskrit(S) = 8

- Number of students learning Sanskrit and Hindi = 8

- Number of students learning English = 26

- Number of students learning Sanskrit = 48

- Number of students learning no language = 24

To Find:

(i) Number of students studying Hindi

Venn diagram:

From the above Venn diagram,

a = Number of students who study only English = 18

b = Number of students who study only Sanskrit

c = Number of students who study only Hindi

d = Number of students learning Hindi and Sanskrit but not English

e = Number of students learning English and Sanskrit but not Hindi

f = Number of students learning Hindi and English but not Sanskrit

g = Number of students learning all the three languages

e + g = Number of students learning English and Sanskrit = 8

= n(E ∩ S)

g + d = Number of students learning Hindi and Sanskrit = 8

= n(H ∩ S)

E = a + e + f + g = Number of students learning English

26 = 18 + 8 + f

f = 26 -26 = 0

Therefore, f = 0

Now,

Number of students learning English but not Hindi = a + e = 23

23 = 18 + e

Therefore, e = 5

Now, e + g = 8

5 + g = 8

Therefore, g = 3

S = b + e + d + g = Number of students studying Sanskrit

48 = b + 5 + 8 (Because, d + g = 8)

b = 48 – 13

Therefore, b = 35 (Number of students studying Sanskrit only)

Also, d + g = 8

d + 3 = 8

Therefore, d = 5

Now,

Number of students studying Hindi only = c

c = 100 – (a + e + b + d + f + g) – 24

= 100 – (18 + 5 + 35 + 5 + 0 + 3) -24

= 100 – 66 – 24

= 100 – 90 = 10

Number of students studying Hindi = c + f + g + d

= 10 + 0 + 3 + 5

= 18

Therefore, number of students studying Hindi = 18

(ii) Number of students studying English and Hindi both

Number of students studying English and Hindi both = f + g

= 0 + 3 = 3

Therefore, Number of students studying English and Hindi both = 3