Question

Consider the following two statements :

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 ${}^{9}C_3$.

Then which of one of the following choices is correct?

• A. Both Statement-$1$ and Statement-$2$are true and Statement-$2$ is a correct explanation for Statement-$1$.
• B. Both Statement-$1$ and Statement-$2$are true and Statement-$2$ is not a correct explanation for Statement-$1$
• C. Statement-$1$ is true but Statement-$2$ is false.
• D. Both Statement-$1$ and Statement-$2$ are true.

Answer 1

The correct option is A

Both Statement-$1$ and Statement-$2$are true and Statement-$2$ is a correct explanation for Statement-$1$.

Explanation for the correct option :

Step-1 : (Formula to be used)

We know that the number of ways of distributing $n$identical balls in $r\le n$ distinct boxes such that no box is empty is same as the number ways of choosing $(r-1)$ places from $\left(n-1\right)$ different places.

Now, the number of ways of choosing any $(r-1)$ places from $\left(n-1\right)$ is same as the number of ways of choosing any $(r-1)$ objects from a collection of $\left(n-1\right)$ distinct objects i.e. ${}^{n-1}C_{r-1}$.

Step-2 : (Apply the above formula)

Here, in this question $n=10$ and $r=4$. So, the number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is same as the number of choosing any $(4-1)=3$ places from $(10-1)=9$ different places which is equal to ${}^{9}C_3$.

Thus both Statement-$1$ and Statement-2 are true and Statement-2 is a correct explanation for Satement-$1$

Hence, option (A) is the correct answer.