Question on permutation and combination

Question

Given answer: B, Please explain me how to solve it.

Answer 1

Ans : (B)

Let x1,x2,x3,x4 represents 3 different triangles 1,2,3,4.

Hence this problem is identical to distributing 8 balls among x1,x2,x3,x4 such that each will get alt least one ball

i.e  x1+x2+x3 +x4  =  8 where x1 ≥ 1; x2 ≥ 1;x3 ≥ 1;x4 ≥ 1;

 = >  (x1-1) + (x2-1) + (x3-1) +(x4 -1) = 4

Now let p = x1-1; q= x2-1;r= x3-1;s=x4-1;

Hence given eqn reduces to :  p+q+r+s = 4 where p,q,r,s ≥ 0

solution to this problem  =  ^(n+r-1)C_r  = ^{(4+4 - 1)}C₄ = ⁷C₄  = 35

Answer 2

let x1 x2 x3 x4 be  no of balls in triangle 1,2,3,4

x1+x2+x3+x4=8

now its given x1>=1 x2>=1 x3>=1 x4>=1

so we get 4+4-1C4=35

option b