Question

In the given figure, what is the maximum number of different ways in which 8 identical balls can be placed in the small



triangles 1,2,3 and 4 such that triangle contains at least one ball?

• A. 32
• B. 35
• C. 44
• D. 56

Answer 1

The correct option is B 35

If you look at it closely, It's same as the case of putting 8 identical balls in 4 different boxes which can be written as:
a + b + c + d = 8, where a, b, c, d represent 4 different boxes or triangles.
Now, it's given that each has to be allotted at least one ball. Therefore, the equation can be further written as:
a + b + c + d = 4 ( after allotting 1 ball each)
Hence, number of ways = ${=}^7{C}_3=35$

Alternatively,
The ways of placing the balls would be 5, 1, 1, 1 $(\frac{4!}{3!}=4ways)$;
4,2,1, 1 $(\frac{4!}{2!}=12ways)$;
3, 3, 1, 1 $(\frac{4!}{2!}\times 2!=6ways)$;
3, 2, 2, 1 $(\frac{4!}{2!}=12ways)$ and 2, 2, 2, 2 (1 way).
Total number of ways = 4 + 12 + 6 + 12 + 1 = 35 ways.