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1
A home alarm system randomly assigns a five-character code for each customer. The code will not repeat a character. The characters are 1, 2, 3, 4, 5, E, M, T, G, Y, and R. What is the total number of codes that can be randomly assigned?
2

Four students out of a class of 18 are being selected to represent the class at the school-wide spelling bee. How many different groups of 4 can be selected?

3
Twelve runners are competing in a race. In how many different ways can the first, second, and third place trophies be awarded?

4

A professor lets his students pick 3 out of 8 assignments to complete. How many combinations of the 3 assignments are possible?
5

Seven runners are competing in a race. In how many different orders can the runners cross the finish line?
6
Skylar is arranging 8 objects in a row on a shelf. In how many different ways can she arrange the objects?
7

A home alarm system randomly assigns a six-character code for each customer. The code will not repeat a character. The characters are 1, 2, 3, 4, 5, M, T, G, Y, or R. What is the total number of codes that can be randomly assigned?

There are 11 items in the set {1, 2, 3, 4, 5, E, M, T, G, Y, R}. So we have 11 choices for the first slot, 11-1 = 10 choices for the second, 10-1 = 9 choices for the third, etc etc until all five slots are filled up. We count down by 1 because we cannot repeat whatever is chosen for a previous slot. Multiply out these values to get 11*10*9*8*7 = 55440. Alternatively, you can use the nPr formula with n = 11 and r = 5 to get the same answer. We use the permutation formula because order matters.

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Problem 2

Answer: 3060

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Explanation:

Unlike problem 1, order does not matter. All that matters is that we have the same students grouped together to form the unique groups. For example, ABCD is the same as ABDC where the letters represent the student names put in shorthand format.

We have 18 choices for the first slot, 17 for the second, 16 for the third, and 15 for the fourth slot. Multiply the values out: 18*17*16*15 = 73440

This would be the answer if order mattered; however, it does not. We have 4! = 4*3*2*1 = 24 ways to arrange any one single group of four people. This means that the value 73440 is too large by a factor of 24. It is 24 times bigger than the true answer. To get the true answer, we divide by 24 to get 73440/24 = 3060

You can use the nCr combination formula with n = 18 and r = 4 to get the same answer. Order does not matter with combinations (in contrast to permutations).

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Problem 3

Answer: 1320

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Explanation:

We have three slots: A, B, C for first place, second place, third place

There are 12 choices for slot A, 11 for B, 10 for C. So, 12*11*10 = 1320

You can use the nPr permutation formula with n = 12 and r = 3 (order matters; similar to problem 1)