Perm Comb

sumithshah wrote:From 4 oranges, 3bananas and 2 apples, how many selections of 5 pieces of fruit can be made, talking at least one of each kind?


OA 98

What I tried to do was 1 from 4 orange 4c1, 1 from 3 bananas, 3c1, one from 2 apples, 2c1 * ( 6c2) [ since there are 2 places left and 6 fruits left.

so basically 4c1*3c1*2c1*6c2 = 360

Any ideas where I went wrong?

Firstly, the answer cannot be 360.

Total combinations without any restrictions = 9C5 = 126
Therefore, if you apply condition such as the one above the total combination will be less than 126.

Selections with at least 1 of each of the fruit = at least 1 + at least 2

or

Total combinations - selection consisting of exactly 2 fruit. [here exactly 1 fruit cannot be possible since we have to make 5 selections, and maximum number of fruit of one type is 4]

Bananas + Oranges = 3+4 = 7 = 7C5 = 21
Oranges + Apples = 3+2 = 5 = 5C5 = 1
Apples + Bananas = 4+2 = 6 = 6C5 = 6


21+1+6 = 28

total combinations = 126

Selections with at least 1 of each of the fruit = 126 - 28 = 98

Hope this helps.