Permutations and Combinations Class 11: notes and important questions

Permutations and Combinations: revision notes

1. Fundamental principle of counting

If one event can happen in \(m\) ways and, after it, another in \(n\) ways, the two together can happen in \(mn\) ways (multiplication principle). Use "and" \(\to\) multiply; "or" (cases that cannot both happen) \(\to\) add.

2. Factorial

\(n! = 1 \times 2 \times \cdots \times n\), \(0! = 1\), \(n! = n \times (n - 1)!\). Simplify quotients before multiplying: \(\dfrac{8!}{6!} = 8 \times 7 = 56\).

3. Permutations (order matters)

• Arrangements of \(r\) of \(n\) different objects: \({}^nP_r = \dfrac{n!}{(n - r)!}\); all \(n\): \(n!\).
• With repetition allowed: \(n^r\).
• \(n\) objects with \(p\) alike of one kind, \(q\) alike of another, ...: \(\dfrac{n!}{p!\,q!\cdots}\).
• Objects that must stay together: treat them as one block, then arrange inside the block.

4. Combinations (order does not matter)

Selections of \(r\) from \(n\) different objects: \({}^nC_r = \dfrac{n!}{r!(n - r)!} = \dfrac{{}^nP_r}{r!}\). Properties: \({}^nC_r = {}^nC_{n-r}\); \({}^nC_r + {}^nC_{r-1} = {}^{n+1}C_r\); if \({}^nC_a = {}^nC_b\) then \(a = b\) or \(a + b = n\).

5. Choosing the method

Arranging, ordering, forming numbers or words \(\to\) permutations. Choosing a team, committee or group \(\to\) combinations. Mixed problems: first select (combinations), then arrange (permutations). Deal with restrictions (a digit that cannot be first, people who must be included) first.

Worked example 1

How many 3-digit numbers with distinct digits can be formed from \(1, 2, 3, 4, 5\)?

\({}^5P_3 = 5 \times 4 \times 3 = 60\).

Worked example 2

In how many ways can a committee of \(3\) be chosen from \(9\) people?

\({}^9C_3 = \dfrac{9 \times 8 \times 7}{6} = 84\).

Worked example 3

How many arrangements are there of the letters of LEVEL?

\(\dfrac{5!}{2!\,2!} = 30\) (two L's, two E's).

Common errors

• Using combinations when order matters (or the reverse).
• Letting \(0\) be the first digit of a number.
• Forgetting to divide by \(p!\) for repeated letters, or to multiply by the arrangements inside a block.
• Adding when the principle needs multiplying ("and"), or multiplying separate cases ("or").

Exam tips

• Draw boxes for the positions and fill in the number of choices for each, restricted positions first.
• Write the formula with numbers, e.g. \({}^{10}C_3 = \dfrac{10 \times 9 \times 8}{3 \times 2 \times 1}\), then simplify.