There are 10 Republicans, 8 Democrats and 2 Independent legislators eligible for committee membership.
How many 5-member committees exist which include at least two Republicans?
My Work :
$(\;All\; 5-member\; committees\;)\;-\;(\;committiees\; with \;no\; Republicans\;+\;committees\; with\; one \;Republican \;)$
Explanation: There are $C(20,5)$ 5-member committees which will be subtracted from sum of two numbers in order to achieve the answer; C(10,5) is number of 5-member committees which include no Republicans (I've only counted Democrats and Independents ($8+2$)then found number of 5-member committees ) and C(10,4) is number of 5-member committees with one Republican (Like before I've excluded Republicans but counted 4-member committees then added one Republican to each subset)
It's just said $C(10,2)C(18,3)$