© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Understand the Problem
We want to distribute 10 identical balls into 4 distinct boxes such that no box remains empty (Statement – 1). In addition, we are given a second statement (Statement – 2) regarding choosing any 3 places out of 9 different places. We must analyze the validity of both statements and see if Statement – 2 explains Statement – 1.
Step 2: Represent the Distribution Mathematically
Let the number of balls in boxes A, B, C, and D be $X_A$, $X_B$, $X_C$, and $X_D$ respectively. Since the boxes are distinct but the balls are identical, and no box can be empty, we have:
$X_A \ge 1,\; X_B \ge 1,\; X_C \ge 1,\; X_D \ge 1 \\
\text{and} \\
X_A + X_B + X_C + X_D = 10.$
To handle the condition "$\ge 1$", we define new variables:
$t_A = X_A - 1,\; t_B = X_B - 1,\; t_C = X_C - 1,\; t_D = X_D - 1.$
Each $t_i$ ($i = A, B, C, D$) satisfies $t_i \ge 0$.
Step 3: Transform the Problem into a Standard Form
Substituting $X_i = t_i + 1$ into the total:
$X_A + X_B + X_C + X_D = 10 \\
\Rightarrow (t_A + 1) + (t_B + 1) + (t_C + 1) + (t_D + 1) = 10 \\
\Rightarrow t_A + t_B + t_C + t_D = 10 - 4 = 6.
Thus, we need to count the nonnegative integer solutions of
$t_A + t_B + t_C + t_D = 6$.
Step 4: Apply the Combinatorial Formula
The number of nonnegative integer solutions to
$t_A + t_B + t_C + t_D = 6$
is given by the formula for distributing $n$ identical items among $r$ distinct containers (where each container can have zero or more items):
Number of ways $= \binom{n + r - 1}{r - 1} = \binom{6 + 4 - 1}{4 - 1} = \binom{9}{3}.$
Hence, Statement – 1 is correct since it matches $\binom{9}{3}$.
Step 5: Analyze Statement – 2
Statement – 2 says the number of ways of choosing any 3 places from 9 different places is $\binom{9}{3}$. This is indeed true because the number of ways to choose 3 distinct elements out of 9 distinct elements is represented by the same binomial coefficient $\binom{9}{3}$.
However, this fact (choosing 3 places out of 9) does not serve as the explanation for the distribution of balls. The two results share the same combinatorial number $\binom{9}{3}$ but arise from different contexts, so Statement – 2, although true, is not a correct explanation for Statement – 1.
Step 6: Conclude the Correct Option
Both statements are true, but Statement – 2 does not explain how to distribute identical balls in distinct boxes. Therefore, the correct answer is:
“Statement – 1 is true, Statement – 2 is true, Statement – 2 is not a correct explanation for Statement – 1.”
