A sandwich shop sells 12 different kinds of sandwiches.
1. You want to buy 5 different sandwiches to bring home to your family. How many ways are there to choose the 5 sandwiches that you will buy?
Would it be 12P5 which is equal to 12*11*10*9*8 = 95,040 ways.
2. Once you get home, you have to distribute the five different sandwiches to your four family members and yourself. How many ways are there for you to do this?
Would this be 5*4*3*2*1 or 5P5 = 120 different ways.
The first one is C(12,5) or 12C5 in your notation
Does it really matter in which order he buys the lunches? NO, so you have a combination
and your answer is 792
but in 2) it does matter which person gets which lunch, so you are correct in 5! or 120 ways
1. To find the number of ways to choose 5 sandwiches out of 12, we use the concept of combinations. The formula for the number of ways to choose r items out of n items without replacement is given by nCr = n! / ((n-r)! * r!), where n is the total number of items and r is the number of items to be chosen.
In this case, n = 12 (the total number of sandwiches available) and r = 5 (the number of sandwiches you want to buy). Plugging these values into the formula, we get:
12C5 = 12! / ((12-5)! * 5!)
= 12! / (7! * 5!)
= (12 * 11 * 10 * 9 * 8) / (5 * 4 * 3 * 2 * 1)
= 95,040
So, there are 95,040 ways to choose 5 different sandwiches to bring home to your family.
2. To find the number of ways to distribute the 5 sandwiches to 4 family members and yourself, we use the concept of permutations. The formula for the number of ways to arrange r items out of n distinct items is given by nPr = n! / (n-r)!, where n is the total number of items and r is the number of items to be arranged.
In this case, n = 5 (the number of sandwiches you have) and r = 5 (the number of family members including yourself). Plugging these values into the formula, we get:
5P5 = 5! / (5-5)!
= 5! / 0!
= (5 * 4 * 3 * 2 * 1) / 1
= 120
So, there are 120 different ways to distribute the 5 different sandwiches to your 4 family members and yourself.