probablility ( combination) I need to know how the formula is done with this question?

Question

A bag of beach glass contains 2 white piece of glass and 3 blue pieces of glass and 4 green pieces of glass. how many ways can three pieces of glass be drawn from the bag if at least 1 piece of blue glass is drawn from the bag?

 

C (n,r) =n

            r! (n-r)!

Answer 1

The formula is n!/(r!(n-r)!) for combinations. n = 9 and r = 3. If there were no restrictions it would be

9x8x7x6x5x4x3x2x1/(3x2x1x6x5x4x3x2x1) and when you cross off the duplicates in the numerator and denominator you get a total of 84.

 

The easiest way to figure the ways with the blue glass is to figure out the ways without the blue glass and then subtract. I like to do this manually. I write

 

WW GGGG to represent the two pieces of white and 4 pieces of green glass. You can count the possibilities starting with two whites and one green, or WWG and you can see that there are 4 ways to pick that total from the whites and greens. Next think about WGG. Starting with white no. 1 and taking two of the 4 green ones, there are 6 ways for each white piece or a total of 12. Finally for three greens, there are 3 ways to pick those. The total of Not P without a blue one is 12 + 4 + 3 or 19. Therefore the total of P is 84 - 19 = 65.

 

Tough problem!  Dorene O.