Statement - 1 : The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is \({ }^{9} \mathrm{C}_{3}\) Statement -2 : The number of ways of choosing any 3 places from 9 different places is \({ }^{9} \mathrm{C}_{3}\).

(D) Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for Statement -1

(A) Statement -1 is true, Statement -2 is true; Statement -2 is not a correct explanation for Statement - 1

(B) Statement -1 is true, Statement- 2 is false.

(C) Statement -1 is false, Statement- 2 is true.

JEE Main · 2011 · 10 Apr 2011 · Offline · Permutations and Combinations

Statement - 1 : The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is

^{9} C_{3}

Statement -2 : The number of ways of choosing any 3 places from 9 different places is

^{9} C_{3}

(A)Statement -1 is true, Statement -2 is true; Statement -2 is not a correct explanation for Statement - 1
(B)Statement -1 is true, Statement- 2 is false.
(C)Statement -1 is false, Statement- 2 is true.
(D)Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for Statement -1✓ Correct

Answer: (D)

The number of ways to put

10

identical balls in

4

distinct boxes with none empty is, by stars and bars,

(4 - 1 10 - 1) = (3 9)

, so the first statement is true. The number of ways to pick

3

places out of

9

(3 9)

, so the second is true as well. The stars-and-bars count is obtained precisely by choosing the

3

divider positions out of

9

, so the second statement is the correct explanation of the first.

JEE Main · 2011 · 10 Apr 2011 · Offline · Single correct option