Combinations. Welcome to Oski\'s Seven Flavors Ice Cream Shop. Your bowl holds f
ID: 3298685 • Letter: C
Question
Combinations. Welcome to Oski's Seven Flavors Ice Cream Shop. Your bowl holds four scoops and we want to know how many ways we can fill it from the seven flavors under the following rules: a) We allow duplication and care about the order that the scoops go in the bowl. b) We don't allow duplication and care about the order that the scoops go in the bowl. c) We don't allow duplication and don't care about the order that the scoops go in the bowl. d) We allow duplication and don't care about the order that the scoops go in the bowl.Explanation / Answer
a) So we are allowing duplicates here but we do care about the order of the flavours. This will be done using the following cases:
Therefore the total number of ways this could be done is: 7 + 168 + 126 + 1260 + 840 = 2401
Therefore 2401 is the number of ways it can be done
b) Here we are given that duplication of flavors is not allowed. Therefore we need to only consider the case 5 in the previous part that is when all 4 flavors are different and they could be arranged in 4! permutations. Therefore the required number of ways is 840 ( from case 5 of previous part )
Therefore 840 ways
c) Here we are given that we dont allow duplication and also we dont care about the order. Therefore we just have to find number of ways of choosing 4 different flavors from the 7 available ones. This is computed as: 7c4 = 35
Therefore 35 ways
d) Here we are given that we allow duplication but dont care about the order of scoops. Again we can consider those same 5 cases in part a) but without permutations. This could be done as:
Therefore the total number of ways this could be done is : 7 + 42 + 21 + 105 + 35 = 210
Therefore in 210 ways