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 ${}^9{C_3}$.
$\left( a \right)$ Statement 1 is true and statement 2 is false.
$\left( b \right)$ Statement 1 is false and statement 2 is true.
$\left( c \right)$ Statement 1 is true and statement 2 is true, statement 2 is a correct explanation for statement 1.
$\left( d \right)$ Statement 1 is true and statement 2 is true, statement 2 is not a correct explanation for statement 1.

Answer 1

In this particular question use the concept that to distribute n number of objects in r number of boxes such that any box take maximum number of books is given as ${}^{n + r - 1}{C_{r - 1}}$, so use these concepts to reach the solution of the question.

Complete step by step answer:
Statement 1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is ${}^9{C_3}$.
So we have 10 identical balls and 4 distinct boxes.
Now we have to find out the number of ways such that no box is empty.
So first distribute one balls to each box, as there are 4 distinct boxes, so the remaining balls = (10 - 4) = 6
Now these 6 balls can go in any box which is not dependent on how much is gone in a particular box (i.e. any box can take a maximum number of balls or some balls or no balls).
So the number of ways to distribute 6 identical balls in 4 distinct boxes such that any box take maximum number of balls is given as, ${}^{n + r - 1}{C_{r - 1}}$, where, n is the total number of balls and r is the number of distinct boxes.
So, n = 6 and r = 4
Now substitute the values we have,
$\Rightarrow {}^{6 + 4 - 1}{C_{4 - 1}} = {}^9{C_3}$
Therefore, statement 1 is true.
Statement 2: The number of ways of choosing any 3 places from 9 different places is ${}^9{C_3}$.
As we all know that the number of ways to select r objects from n objects is given as, ${}^n{C_r}$
So the number of ways of choosing any 3 places from 9 different places is ${}^9{C_3}$.
So statement 2 is also true.
But as we see there is no relation between statement 1 and statement 2.
Statement 1 is true and statement 2 is true, statement 2 is not a correct explanation for statement 1.
So this is the required answer.

So, the correct answer is “Option d”.

Note: Whenever we face such types of questions the key concept we have to remember is that we always recall that the number of ways to select r objects from n objects is given as, ${}^n{C_r}$. So first solve statement 1 then solve statement 2, if both are true, then check whether there is a relation between them or not.