Common Core Math
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Exercises: Permutations

Work through each section in order. Show your work where indicated.

Grade 10·18 problems·~25 min·ACT Math·topic·act-sp-counting-perm
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A

Recall / Warm-Up

These problems review prerequisite skills.

1.

A student has 4 shirts and 3 pairs of pants. Using the fundamental counting principle, how many different outfits can the student make?

2.

What is the value of the expression n!n! when n=0n = 0?

3.

Which scenario requires counting ordered arrangements rather than unordered groups?

B

Fluency Practice

Use the permutation formula or factorial simplification to compute each value.

1.

Evaluate 7!5!\frac{7!}{5!}.

2.

What is P(6,2)P(6, 2)?

3.

Compute P(9,3)P(9, 3).

4.

Compute P(5,5)P(5, 5).

C

Mixed Practice

These problems test the same skills in different ways.

1.

A shelf has room for exactly 4 books chosen from a collection of 7 different books. How many ways can the 4 books be arranged on the shelf?

2.

A lock requires a 3-digit code using the digits 0 through 9, with no digit repeated. How many possible codes are there?

3.

From 12 runners in a race, how many different ways can gold, silver, and bronze medals be awarded?

4.

Which of the following situations does NOT require permutations?

D

Word Problems

Read each problem carefully. Determine whether order matters, then solve.

1.

A band director must arrange 6 musicians in a line on stage.

How many different stage arrangements are possible?

2.

A phone's unlock pattern requires selecting 4 distinct dots from a grid of 9 dots in a specific sequence.

How many different unlock patterns are possible?

3.

A teacher wants to display 5 different student paintings in a row along a hallway wall, chosen from 8 paintings submitted.

How many different displays are possible?

E

Error Analysis

Each problem shows a student's work that contains an error. Find and explain the mistake.

1.

Kira was asked: "From 10 athletes, how many ways can a coach assign 1st, 2nd, and 3rd place?"

Kira's work:
C(10,3)=10!3!⋅7!=7206=120C(10, 3) = \frac{10!}{3! \cdot 7!} = \frac{720}{6} = 120

What mistake did Kira make?

2.

Marcus was asked: "How many ways can all 4 students in a group be arranged in a line?"

Marcus's work:
P(4,4)=4!(4−4)!=4!0!=240=undefinedP(4, 4) = \frac{4!}{(4-4)!} = \frac{4!}{0!} = \frac{24}{0} = \text{undefined}

Marcus concluded the problem has no solution.

What mistake did Marcus make?

F

Challenge

These are bonus problems that require multi-step reasoning.

1.

A class of 15 students elects a president, a vice president, and a secretary. The president then appoints 2 of the remaining 12 students as co-chairs of a committee (where both co-chairs have equal roles). How many ways can all of these positions be filled?

2.

How many 5-letter "words" (any arrangement of letters, not necessarily real words) can be formed from the 26 letters of the alphabet if no letter may be used more than once?

0 of 18 answered