Math 30-1

Permutations, combinations and the binomial theorem

Twenty questions, all on the easier end. Every one has a hint if you want it. Check your answers as many times as you like — you are told what is right, never what the answer is.

Your name (optional)

Only used to say hello in your score. It stays in this browser and is never sent anywhere.

Fundamental counting principle

Key idea. If one decision can be made in m ways and a second in n ways, the two together can be made in m × n ways. Draw a box for each decision, write the number of choices in it, and multiply.

Question 1

Maya has 3 shirts and 4 pairs of pants. How many different outfits (1 shirt and 1 pair of pants) can she make?

a)7b)12c)24d)34

Show a hint

When you make one choice AND another, multiply.

Question 2

A restaurant special includes 1 appetizer, 1 main course, and 1 dessert. There are 2 appetizers, 5 main courses, and 3 desserts. How many different meals are possible?

a)10b)15c)30d)60

Show a hint

Multiply the number of options at each stage.

Question 3

A coin is flipped 4 times. How many different sequences of heads and tails are possible?

a)4b)8c)16d)24

Show a hint

Each flip has 2 possible outcomes.

Question 4

A lock code uses 3 digits (0–9), and digits may be repeated. How many codes are possible?

a)30b)720c)999d)1000

Show a hint

How many digits are there from 0 to 9?

Question 5

A 4-digit PIN is made using only the digits 1, 2, 3, 4, and 5, with no digit repeated. How many PINs are possible?

a)20b)24c)120d)625

Show a hint

After you use a digit, there’s one fewer choice for the next spot.

Permutations

Key idea. Order matters. ₙPᵣ = n! ÷ (n − r)!. To arrange n objects where some repeat, divide n! by the factorial of each repeat count.

Question 6

What is the value of 5! ?

a)25b)60c)120d)720

Show a hint

Multiply 5 by every whole number below it, down to 1.

Question 7

In how many ways can 4 different books be arranged in a row on a shelf?

a)4b)16c)24d)256

Show a hint

Order matters and every book is used.

Question 8

Evaluate ₈P₂.

a)16b)28c)56d)64

Show a hint

Use n! ÷ (n − r)!.

Question 9

A club with 6 members must choose a president and a vice-president. How many ways can this be done?

a)12b)15c)30d)36

Show a hint

Does it matter who gets which position?

Question 10

How many different arrangements can be made using all the letters of the word BOOK?

a)4b)6c)12d)24

Show a hint

Divide by the factorial of the number of repeated letters.

Combinations

Key idea. Order doesn’t matter. ₙCᵣ = n! ÷ [(n − r)! r!]. A handy link: ₙCᵣ = ₙPᵣ ÷ r!, because each group can be ordered r! ways.

Question 11

Evaluate ₅C₂.

a)7b)10c)20d)25

Show a hint

Use n! ÷ [(n − r)! r!].

Question 12

Which situation should be solved using a combination instead of a permutation?

a)Awarding gold, silver, and bronze medals to 3 of 10 runnersb)Choosing 3 students from a class of 25 to attend a field tripc)Creating a 3-letter passwordd)Arranging 3 photos in a row on a wall

Show a hint

Look for the situation where order doesn’t matter.

Question 13

A pizza shop offers 6 toppings. How many different pizzas with exactly 3 different toppings can be made?

a)18b)20c)120d)216

Show a hint

Is pepperoni-mushroom-onion different from onion-mushroom-pepperoni?

Question 14

A committee of 2 people is chosen from a group of 10. How many different committees are possible?

a)20b)45c)90d)100

Show a hint

Order doesn’t matter on a committee.

Question 15

At a meeting, each of 6 people shakes hands once with every other person. How many handshakes occur?

a)12b)15c)30d)36

Show a hint

A handshake between A and B is the same as between B and A.

Binomial theorem

Key idea. (a + b)ⁿ has n + 1 terms. The coefficients are row n of Pascal’s triangle, which are also ₙC₀, ₙC₁, … , ₙCₙ. The general term is ₙCₖ aⁿ⁻ᵏ bᵏ.

Question 16

How many terms are in the expansion of (x + y)⁶?

a)5b)6c)7d)12

Show a hint

Try (x + y)² first. How many terms does it have?

Question 17

Which row of Pascal’s triangle gives the coefficients in the expansion of (a + b)³?

a)1 2 1b)1 3 3 1c)1 4 6 4 1d)1 3 1

Show a hint

The expansion will have 4 terms.

Question 18

What is the expansion of (x + 1)²?

a)x² + 1b)x² + x + 1c)x² + 2x + 1d)2x + 2

Show a hint

Don’t forget the middle term.

Question 19

In the expansion of (a + b)⁴, what is the coefficient of the middle term, a²b²?

a)2b)4c)6d)8

Show a hint

Use the row of Pascal’s triangle that starts 1, 4, …

Question 20

What is the first term in the expansion of (2x + 3)³?

a)2x³b)6x³c)8x³d)27

Show a hint

The whole term 2x gets raised to the power of 3.

Teachers and parents: the illustrated answer key is behind a code.

0 of 20 answered