Counting, Permutations & the Binomial Theorem — Drills
Counting, Permutations & the Binomial Theorem has 6 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills 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)!.
Factorial
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Evaluate 3!.
Worked answer
6
1 · 2 · 3 = 6
Problem 2
Evaluate 5!.
Worked answer
120
1 · 2 · 3 · 4 · 5 = 120
Combination
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Evaluate C(7,2).
Worked answer
21
7!2!5! = 21
Problem 2
Evaluate C(9,2).
Worked answer
36
9!2!7! = 36
Permutation
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Evaluate P(7,2).
Worked answer
42
7 × 6 = 42
Problem 2
Evaluate P(9,2).
Worked answer
72
9 × 8 = 72
Counting
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A meal is one of 5 starters, one of 5 mains and one of 5 puddings. How many meals?
Worked answer
125
5 × 5 × 5 = 125
Problem 2
A meal is one of 6 starters, one of 4 mains and one of 3 puddings. How many meals?
Worked answer
72
6 × 4 × 3 = 72
Binomial
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the coefficient of x3 in (x+3)5.
Worked answer
90
Need r=2, so C(5,2) · 32
= 10 × 9 = 90
Problem 2
Find the coefficient of x4 in (x+2)6.
Worked answer
60
Need r=2, so C(6,2) · 22
= 15 × 4 = 60
Symmetry
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
C(9,6) equals C(9,j) for which other j?
Worked answer
3
C(n,r) = C(n,n-r)
9-6 = 3
Problem 2
C(11,6) equals C(11,j) for which other j?
Worked answer
5
C(n,r) = C(n,n-r)
11-6 = 5