Jacob wants to make a sandwich with 3 kinds of vegetables on it, but there are 6 vegetables to choose from in his refrigerator. In how many sequences can he layer 3 of them on his sandwich?
Question
Jacob wants to make a sandwich with 3 kinds of vegetables on it, but there are 6 vegetables to choose from in his refrigerator. In how many sequences can he layer 3 of them on his sandwich?
Solution
To solve this problem, we need to use the concept of permutations in combinatorics. A permutation is an arrangement of objects in a specific order. The number of permutations of n objects taken r at a time is given by the formula:
P(n, r) = n! / (n-r)!
where "!" denotes factorial, which is the product of all positive integers up to that number.
In this case, Jacob has 6 vegetables to choose from (n = 6) and he wants to choose 3 of them (r = 3).
So, the number of sequences (permutations) in which he can layer 3 vegetables on his sandwich is:
P(6, 3) = 6! / (6-3)!
= 654321 / 32*1
= 654
= 120 sequences.
So, Jacob can make his sandwich in 120 different sequences of vegetables.
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