How many samples contain at least 1 red ball?
How many samples contain exactly 4 red balls and exactly 3 black balls?
How many samples contain exactly 5 green balls or exactly 3 black balls?
How many samples contain exactly 4 red balls and exactly 3 black balls?
How many samples contain exactly 5 green balls or exactly 3 black balls?
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Total combinations = 25 c 9 = 2042975
At least one red = total of all - (those with no red)
no red → all green and black = 15c9 = 5005
The rest (2042975 - 5005) have at least one red.
Exactly 4 red and exactly 3 black:
10c4 ways to choose the red
9c3 ways to choose the black
6c2 ways to choose the remaining 2 green
Multiply those together to get the answer.
5 green can be accompanied by any 4 of the others: 6c5 * 19c4
3 black can be accompanied by any 6 of the others: 9c3 * 16c6
But those count
the combos with 5 green and 3 black and 1 red twice (once in each category)
so we must subtract one of them:
answer is
6c5 * 19c4 + 9c3 * 16c6 - (6c5 * 9c3 * 10c1)
At least one red = total of all - (those with no red)
no red → all green and black = 15c9 = 5005
The rest (2042975 - 5005) have at least one red.
Exactly 4 red and exactly 3 black:
10c4 ways to choose the red
9c3 ways to choose the black
6c2 ways to choose the remaining 2 green
Multiply those together to get the answer.
5 green can be accompanied by any 4 of the others: 6c5 * 19c4
3 black can be accompanied by any 6 of the others: 9c3 * 16c6
But those count
the combos with 5 green and 3 black and 1 red twice (once in each category)
so we must subtract one of them:
answer is
6c5 * 19c4 + 9c3 * 16c6 - (6c5 * 9c3 * 10c1)