Counting, Permutations & the Binomial Theorem — Practice
Expanding a bracket and counting arrangements turn out to be the same question.
What these questions cover
These questions use 4 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Independent — The first event does not change the second — like rolling a die twice.
- Coefficient — The number multiplying a variable — the 3 in 3n.
- Permutation — An arrangement where order matters. P(n,r) = n!(n-r)!.
- Combination — A selection where order does not matter. C(n,r) = n!r!(n-r)!.
Problem 1
Evaluate 5!.
Worked answer
120
1 · 2 · 3 · 4 · 5 = 120
Problem 2
Evaluate 0!.
Worked answer
1
0! = 1 by definition
Problem 3
A menu has 4 starters and 3 mains. How many two-course meals?
Worked answer
12
4 × 3 = 12
Problem 4
Evaluate C(5,2).
Worked answer
10
5 × 42 = 10
Problem 5
Evaluate P(5,2).
Worked answer
20
5 × 4 = 20
Problem 6
A committee of 3 is chosen from 8 people. Which applies?
- A combination
- A permutation
- Neither
- Both give the same
Worked answer
A combination
No, so order does not matter
Problem 7
Gold, silver and bronze are awarded to 3 of 8 runners. Which applies?
- A permutation
- A combination
- Neither
- Both give the same
Worked answer
A permutation
Yes, so order matters
Problem 8
How many committees of 3 can be chosen from 8 people?
Worked answer
56
8 × 7 × 66 = 56
Problem 9
How many ways to award gold, silver and bronze among 8 runners?
Worked answer
336
8 × 7 × 6 = 336
Problem 10
Give the sum of row 4 of Pascal’s triangle, which is 1, 4, 6, 4, 1.
Worked answer
16
1+4+6+4+1 = 16 = 24
Problem 11
Give the coefficient of a2b3 in the expansion of (a+b)5.
Worked answer
10
5 × 42 = 10
Problem 12
Find the coefficient of x3 in (x+2)5.
Worked answer
40
Need r=2, so C(5,2)22
10 × 4 = 40
Problem 13
Find the coefficient of x2 in (x+3)4.
Worked answer
54
6 × 9 = 54
Problem 14
A bag has 5 red and 3 blue marbles. Two are drawn at once. What is the probability both are red? Give a fraction in lowest terms.
Worked answer
5/14
C(5,2)C(8,2) = 1028
= 514
Problem 15
C(9,7) equals C(9,k) for which smaller k?
Worked answer
2
C(n,r) = C(n,n-r)
9-7 = 2
Problem 16
What is the probability of exactly 3 heads in 5 fair coin flips? Give a fraction in lowest terms.
Worked answer
5/16
C(5,3)25 = 1032
= 516