Question

Statement I The number of ways distributing $10$, identical balls in $4$ distinct boxes such that no box is empty, is ${}^9{C}_3$ .

Statement II The number of ways of choosing any $3$ places from $9$ different places, is ${}^9{C}_3$ .

Which is correct?

• A. Statement I is correct, Statement II is correct; Statement II is not a correct explanation for Statement I
• B. Statement I is correct, Statement II is incorrect
• C. Statement I is incorrect, Statement II is correct
• D. Statement I is correct, Statement II is correct; Statement II is a correct explanation for Statement I

Answer 1

The correct option is A

Statement I is correct, Statement II is correct; Statement II is not a correct explanation for Statement I

Explanation for the correct option:

Finding the required number of ways given in the statements:

Suppose the number of ways to distribute $n$ identical objects amongst $r$ persons where each person gets at least one object is equal to the number of ways of selecting $(r-1)$ places out of $(n-1)$ different places, is given by ${}^{n-1}{C}_{r-1}$

Therefore,

According to Statement I,

Number of distinct boxes $r=4.$

Number of identical balls $n=10$

Now, the number of ways will become ${}^{10-1}C_{4-1}={}^{9}C_3$

Therefore, the statement I is correct

Also, for statement II,

The Number of ways will become ${}^{9}C_3$.

Therefore, the statement II is correct.

Hence, both statements are true but Statement II is not the correct explanation of the other statement.

Therefore, the correct answer is option (A).