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We have 5 flavors of ice cream and we can take 3 scoops in each or which can be any of the flavors.
Now as we get the same variant whether we take the 1st flavor first or take the first flavor last, so our order of choosing flavors does not matter. We are only interested in finding the total number of combinations, not the total number of permutation.
Now the formula for filling 3 slots, each with one of the 5 flavors is given by C (n + r -1, r) = [(n+r-1)!] / [r!*(n-1)!]
Here n is 5 and r is 3.
Therefore C (n + r -1, r) = [(n+r-1)!] / [r!*(n-1)!]
=> [5 + 3 - 1)! ] / [3!*(5-1)!]
=> 7! / ( 3! * 4!)
=> 7*6*5/ 3*2
The number of variants possible is 35.
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