Question

be sure to show your work for full credit.

1. how many outfits can mr. jack make out of 4 shirts, 3 pairs of pants, two belts, and one pair of shoes.
2. how many different ways can six students line up?
3. how many 3 member committees can be formed from a club with 15 people?
4. a science quiz has eight multiple choice questions with four choices each. find the total number of ways to answer the questions.
5. there are 16 skaters competing in the competition. how many ways can they win the gold, silver, and bronze medal?
6. there are 23 students in ms. malboeuf’s class. how many different 3 person groups can she make?
7. how many ways can 8 different books be arranged on a shelf?
8. how many distinct words can be formed from the letters in the word multiply?
9. markel, jacob, and jonas stand in a line. how many different ways can they stand in line?
10. a pizza shop has available toppings of anchovies, mushrooms, onions, olives, and peppers. how many different ways can a pizza be made with 3 toppings?

Explanation:

Problem 4

Step1: Identify the principle

Each multiple - choice question has 4 choices, and there are 8 such questions. We use the Fundamental Counting Principle (FCP), which states that if there are \(n_1\) ways to do the first task, \(n_2\) ways to do the second task, \(\cdots\), \(n_k\) ways to do the \(k\) - th task, then the total number of ways to do all \(k\) tasks together is \(n_1\times n_2\times\cdots\times n_k\). Here, for each of the 8 questions (tasks), there are 4 choices (ways to answer).

Step2: Apply the FCP

The number of ways to answer the 8 questions is \(4\times4\times\cdots\times4\) (8 times). Using the formula for exponents, \(a^n\) where \(a = 4\) and \(n=8\), we have \(4^8\).
Calculate \(4^8=4\times4\times4\times4\times4\times4\times4\times4 = 65536\).

Step1: Identify the permutation formula

We want to find the number of ways to award 3 distinct medals (gold, silver, bronze) to 16 skaters. The number of permutations of \(n\) objects taken \(r\) at a time is given by the formula \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 16\) (total number of skaters) and \(r=3\) (number of medals).

Step2: Calculate the permutation

\(P(16,3)=\frac{16!}{(16 - 3)!}=\frac{16!}{13!}\)
Since \(n!=n\times(n - 1)\times\cdots\times1\), we can write \(16! = 16\times15\times14\times13!\)
So, \(\frac{16\times15\times14\times13!}{13!}=16\times15\times14 = 3360\)

Step1: Identify the combination formula

We want to form 3 - person groups from 23 students. The number of combinations of \(n\) objects taken \(r\) at a time is given by the formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 23\) and \(r = 3\).

Step2: Calculate the combination

\(C(23,3)=\frac{23!}{3!(23 - 3)!}=\frac{23!}{3!×20!}\)
Since \(23! = 23\times22\times21\times20!\), we have \(\frac{23\times22\times21\times20!}{3\times2\times1\times20!}\)
Simplify the expression: \(\frac{23\times22\times21}{6}=\frac{23\times11\times21}{3}=23\times11\times7 = 1771\)

Answer:

65536

Problem 5