Assertion :The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is 9C3. Reason: The number of ways of choosing any 3 places from 9 different places is 9C3.
Assertion :The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is 9C3. Reason: The number of ways of choosing any 3 places from 9 different places is 9C3.
The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
First of all, this is a question that uses the "stars and bars" technique. In case you don't know "stars and bars", we can think of the problem as laying out the 10 balls in a row and then building boxes around the balls. Since the two walls "at the end" of the boxes is trivial, we ignore them and look only at the walls that actually divide the balls.
We can denote the balls with a 0 and the walls of the boxes as a 1.
We can have this:
0001000100010
which is 3 balls in three boxes and 1 ball in one box.
And so what we can do is look at the number of ways we can distribute the walls (the 1s). This is a combinations problem, the formula for which is:
we can assign 4 balls and put one each into a box. That leaves 6 balls to be divided amongst the 4 boxes. Since we need 3 box walls to denote the 4 boxes (just like in our example above), we can then find unique combinations of 6 balls and 3 walls across 4 walls (using 3 walls) is:
9C3=84
Hence, both the statements are true and B is the correct explanation for A.
First of all, this is a question that uses the "stars and bars" technique. In case you don't know "stars and bars", we can think of the problem as laying out the 10 balls in a row and then building boxes around the balls. Since the two walls "at the end" of the boxes is trivial, we ignore them and look only at the walls that actually divide the balls.
We can denote the balls with a 0 and the walls of the boxes as a 1.
We can have this:
0001000100010
which is 3 balls in three boxes and 1 ball in one box.
And so what we can do is look at the number of ways we can distribute the walls (the 1s). This is a combinations problem, the formula for which is:
we can assign 4 balls and put one each into a box. That leaves 6 balls to be divided amongst the 4 boxes. Since we need 3 box walls to denote the 4 boxes (just like in our example above), we can then find unique combinations of 6 balls and 3 walls across 4 walls (using 3 walls) is:
9C3=84
Hence, both the statements are true and B is the correct explanation for A.
