Im looking over an exam I took in finite mathmatics recently

I\'m looking over an exam I took in finite mathmatics recently, and I\'m still getting tippred up over this question:

There are 25 marbles, 7 blue , 8 green, and 10 red. How many ways can you choose at least 1 blue, and 1 green, if you must choose 6?

For the life of me I cannot figure this out. How would I solve this?

Solution

you just heed to see how many green there are

you will choose 1 green and you have 8 green

8C1 = 8

now when it says at least you can have 1, or 2 or 3 or 4 or 5 blue

that means

7 C 1 + 8 + 23C4 =88870

7 C 2 + 8 + 22C3=1569

7C 3 + 8 + 21 C2 = 253

7C4 + 8 + 20C1 = 63

7C5 + 8 +19C0= 29

in total we have 90784 different ways