Question

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, Statement -2 both are true But 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

Answer 1

Hint – In order to solve this problem we need to know that the number of ways of distributing n items at r places such that no place is empty is same as the number of ways we have to distribute n-1 items at r-1 places and it is$^{n - 1}{C_{r - 1}}$. Doing this will solve your problem.

Complete step by step solution:
On considering statement – 1:
The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is $^9{C_3}$.
On analysing the above situation we know that it is same as the number of ways of distributing n identical objects among r persons such that each person gets at least one object in same as the number of ways of selecting (r−1) places out of (n−1) different places, that is, $^{n - 1}{C_{r - 1}}$.
So, statement 1 is correct.
On considering statement 2:
The number of ways of choosing any 3 places from 9 different places is $^9{C_3}$.
This statement is very clear that the number of ways of distributing n items at r places such that no place is empty is $^n{C_r}$.
Therefore statement 2 is also correct.
But statement 2 is not the correct explanation of statement 1.
Hence, the correct option is statement A.
Note – When you get to solve such problems we need to take care that if we have to distribute the items or we have to pick up the items then we have to solve the problem accordingly and get the results using the concepts and facts mentioned above. Doing this will solve the problem and will give you the right answer.