Question

Statement 1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is ${}_{}^9C_3^{}$.
Statement 2: The number of ways of choosing any three places from 9 different places is ${}_{}^9C_3^{}$.
A. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is true, Statement-2 is false
D. Statement-1 is false, Statement-2 is true

Answer 1

Hint: This problem requires knowledge of permutations and combinations. We will first find the number of ways to arrange 10 identical balls in 4 distinct boxes such that no box is empty. The formula to be used is-
${}_{}^{n + r - 1}C_{r - 1}^{}$

Complete step-by-step answer:
Then we will verify if the statement is true or not. Then we will find the number of ways to choose 3 places from 9 different places-
${}_{}^nC_r^{} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}$
Then we will check if the given statement is true or not.

We have to find the number of ways to arrange 10 identical balls in 4 distinct boxes such that no box is empty. So, first we will put one ball each in the four boxes, so that no box remains empty. Now, we can arrange the remaining 6 balls in the 4 boxes using the formula-
${}_{}^{n + r - 1}C_{r - 1}^{}$
Substituting n = 6 and r = 4 we get-
${}_{}^{6 + 4 - 1}C_{4 - 1}^{} = {}_{}^9C_3^{}$
Hence, Statement-1 is true.

Now, we will find the number of ways to select 3 places out of 9, which can be done simply using the formula for selecting n things out of r as-
${}_{}^nC_r^{} = {}_{}^9C_3^{}$
Hence, Statement-2 is also true.

Both the statements are true, but we can see that we used different approaches and formulas to compute the number of ways, so we can say that Statement-2 is not a correct explanation for Statement-1. Hence, the correct option is B.

Note: A common misconception here is that students assume the final answer in both the statements is the same so Statement-2 should be a possible explanation for Statement-1, which is wrong. We need to inspect the method and approach used in both the parts, to identify which option should be correct. Also, students should read the statements closely for words like ‘distinct’ and ‘identical’ as they can modify the approach significantly.