Flashcards · 12 cards

Permutation & combination

Say the answer to yourself, then check. Cards you know come back less and less often; cards you don't come back tomorrow.

Know it
${}^nP_r$ and ${}^nC_r$ ?
Answer
${}^nP_r = \dfrac{n!}{(n - r)!}$ (order matters) and ${}^nC_r = \dfrac{n!}{r!\,(n - r)!}$ (order doesn't).
From the lesson: Equations and harder arrangements
Rule
How are ${}^nP_r$ and ${}^nC_r$ related?
Answer
${}^nP_r = {}^nC_r \times r!$ : each selection of $r$ things gives $r!$ arrangements.
Arrangements and selectionsEach selection of r gives r! arrangements: ⁿPᵣ = ⁿCᵣ × r!
From the lesson: Equations and harder arrangements
Know it
Solve ${}^nC_2 = 45$ .
Answer
Write it out: $\dfrac{n(n - 1)}{2} = 45$ , so $n^2 - n - 90 = 0$ and $n = 10$ (not $-9$ ).
From the lesson: Equations and harder arrangements
Rule
Some items must stand together. How do you count the arrangements?
Answer
Glue them into one block. Arrange the blocks and single items, then arrange inside each block, and multiply.
Glue them togetherArrange the units, then arrange inside each block
From the lesson: Equations and harder arrangements
Rule
Arranging letters when some repeat?
Answer
$\dfrac{n!}{p!\,q!\cdots}$ : divide by the factorial of each repeat count (three A's: divide by $3! = 6$ , not 3).
Take the tags offA, A, B: 3! orders with the A's tagged, but each word appears 2! times, so 3!/2! = 3
From the lesson: Equations and harder arrangements
Know it
One place is restricted, such as the last digit being even. Where do you start?
Answer
Fill the restricted place first, then the others.
From the lesson: Equations and harder arrangements
Rule
A committee must have at least 2 women. How do you count?
Answer
Count each case (2 women, 3 women, …) separately, then add.
At least 2 womenCount each case, then add
From the lesson: Committees with conditions and probability
Rule
Splitting $n$ people into $k$ equal, unnamed groups of $m$ ?
Answer
$\dfrac{n!}{(m!)^k\,k!}$ : divide by $k!$ because swapping whole groups gives the same split.
Unnamed pairsA, B, C, D into two pairs: numbering the pairs counts each split 2! times
From the lesson: Committees with conditions and probability
Know it
Counting: when do you multiply and when do you add?
Answer
Choosing from one group and another: multiply. One case or another: add.
From the lesson: Committees with conditions and probability
Know it
A probability from selections?
Answer
$\dfrac{\text{number of selections that fit}}{\text{total number of selections}}$ .
From the lesson: Committees with conditions and probability
Which method?
WAEC 2016 · Paper 2 · Q13 (a)
Two different Mathematics books, 5 different Physics books and 3 different Chemistry books are to be arranged on a shelf. How many arrangements are possible if: (i) books on the same subject must stand together; (ii) only the Physics books must stand together?
How do you keep each subject together?
Answer
Glue each subject into a block: $3!$ ways to order the three blocks, times $2! \times 5! \times 3!$ for the orders inside them.
From the lesson: Equations and harder arrangements Try the question
Which method?
WAEC 2011 · Paper 2 · Q13 (a)
A committee of five is to be formed among 6 Ghanaians, 8 Nigerians and 5 Gambians. In how many ways can the committee be formed if: (i) there is no restriction; (ii) at most 2 Ghanaians are on the committee; (iii) 1 Nigerian is on the committee?
"At most 2 Ghanaians": which cases?
Answer
0, 1 or 2 Ghanaians, with the rest chosen from the 13 others. Count each case and add.
From the lesson: Committees with conditions and probability Try the question