Question

there are 5 oranges,4 mangoes ,6 apples .in how many ways can selection of fruit be made taking at least one of each kind

Answer 1

Answer:

Here a+b = n

Now let a= x+1 & b=y+1

So, x+1+y+1 = n

=> x+y = n-2

0 ≤ x ≤ n-2 & 0 ≤ y ≤ n-2

Total number of +ve integral solutions of a+b = n is same as total number of non -ve integral solution of x+y = n-2 i.e. co-efficient of x^(n-2) in (1-x)^-2

= 2+n-2-1Cn-2 = n-1Cn-2 ways {where nϵ[2,9]}.

=9–1C9–2 + 8–1C8–2 +… +2–1C2–2

= 8C7 + 7C6 + … + 1C0

=8 + 7 + … + 1 =36 ways

I believe this is the shortest solution of this question. We used multinominal theorem.

It's explanation can also be given using Geometric Progression but it would be too complicated to write it over here.