Answer to Question #259739 in Discrete Mathematics for rtf

Question

There are piles of identical red, blue, and green balls where each pile contains at least 10 balls.

(a) In how many ways can 10 balls be selected if exactly one red ball must be selected? (

(b) In how many ways can 10 balls be selected if at least one red ball, at least two blue balls, and at least three green balls must be selected? 

Answer 1

a). If exactly one red ball must be selected, then each of 9 balls left can be green or blue. Since the removal order is not important here, we have 10 possible outcomes(the amount of green balls can be from 0 to 9, and for each situation other places automatically filled with blue balls)

b). If there must be at least one red, at least two blue and at least three green balls, then 6 places are occupied, and we have to find how many ways exists to fill other 4 places. On each of the four place can be either red, blue, or green ball, so there are such possible ways:

0 red balls (5 ways)

1 red ball (4 ways)

2 red balls (3 ways)

3 red balls (2 ways)

4 red balls (1 way)

5 + 4 + 3 + 2 + 1 = 15 ways