Mai is arranging 12 cans of food in a row on a shelf. She has 5 cans of carrots, 1 can of beets, and 6 cans of olives. In how many distinct orders can the cans be arranged if two cans of the same food are considered identical (not distinct)?
Question
Mai is arranging 12 cans of food in a row on a shelf. She has 5 cans of carrots, 1 can of beets, and 6 cans of olives. In how many distinct orders can the cans be arranged if two cans of the same food are considered identical (not distinct)?
Solution
This is a permutation problem involving identical objects. The formula for permutations of n objects where there are n1 of one type, n2 of another, etc. is:
n! / (n1! * n2! * ...)
In this case, we have:
- n = 12 (total number of cans)
- n1 = 5 (number of carrot cans)
- n2 = 1 (number of beet cans)
- n3 = 6 (number of olive cans)
So, we can plug these values into the formula:
12! / (5! * 1! * 6!) = 95040
So, there are 95040 distinct ways to arrange the cans on the shelf.
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