Question

How many ways are there to choose 10 pieces of fruit from oranges, apples, bananas,
and peaches such that you have at most 3 kinds of fruit? Hint: you don't want 4 kinds of
fruit.

Answer 1

number of ways to choose 3 fruits from 4 =⁴C₃ =4

let x1,x2 and x3 are number of fruits of 3 types

therefore x1+x2+x3=10 where x1,x2,x3 >=1

number of solution of above =^10-1C_3-1 =36

number of ways to choose 2 fruits from 4 =⁴C₂ =6

let x1 and x2 are number of fruits of 2 types

x1+x2 =10 where x1,x2 >=1

number of solution of above =^10-1C_2-1 =9

number of ways to choose 1 fruits from 4 =⁴C₁ =4

number of solution for putting 10 in one type of fruit =1

hence total number of ways choose 10 pieces of fruit from oranges, apples, bananas,
and peaches such that you have at most 3 kinds of fruit=4*36+6*9+4*1 =202