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$A_{1}, A_{2}, A_{3}, . . ., A_{30}$ be thirty sets with 5 elements and $B_{1}, B_{2}, . . ., B_{n}$ be $n$ sets each with $3$ elements. If $C$ is a set containing $m$ elements such that each element of $C$ belongs to exactly $10$ of the $A_{i}^{'} s$ and exactly $9$ of the $B_{j}^{'} s$ and $C = 30 \cup i = 1 A_{i} = n \cup j = 1 B_{j}$ , then find the number of positive integral solutions of equations of equation $x_{1} + x_{2} + . . . + x_{15} = 45$

A

^{44} C_{14}

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B

^{44} C_{15}

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C

^{45} C_{14}

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D

^{45} C_{15}

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Solution

The correct option is D $^{44} C_{14}$
The number of positive integral solution of given equation is $=^{45 - 1} C_{15 - 1} =^{44} C_{14}$

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Similar questions

A_{1}, A_{2}, . . . A_{30}

are thirty sets each having 5 elements and

B_{1}, B_{2}, . . ., B_{n}

are n sets each with 3 elements , let

30 \cup i = 1 A_{i} = n \cup j = 1 B_{j} = S

and each element of S belongs to exactly 10 of the

A_{i}^{'} s

and exactly 9 of the

B_{j}^{'} S

. then n is equal to

A_{1}, A_{2}, \dots, A_{30}

are thirty sets each having

5

B_{1}, B_{2}, \dots, B_{n}

n

3

30 ⋃ i = 1 A_{i} = n ⋃ j = 1 B_{j} = S

and each element of

S

belongs to exactly

10

A_{i}^{'} s

9

B_{j}^{'} s .

n

A_{1}, A_{2}, . . ., A_{30}

are thirty sets each having 5 elements and

B_{1}, B_{2}, . . ., B_{n}

are n sets each with 3 elements. Let

\cup_{i = 1}^{30} A_{i} = \cup_{j = 1}^{n} B_{j} = S

and each element of S belong to exactly 10 of the

A_{i}' s

and exactly 9 of the

B_{j}' s

, then n is equal to

(a) 15 (b) 3 (c) 45 (d) 35

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Suppose $A_{1}, A_{2}, . . ., A_{30}$ are thirty sets, each with five elements and $B_{1}, B_{2}, . . ., B_{30}$ are $n$ sets ecah with three elements. Let $30 ⋃ i = 1 A_{i} = n ⋃ j = 1 B_{j} = S$

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