Flashcards · 9 cards

Counting

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

Rule
How do you count choices made one after another?
Answer
Multiply the number of choices for each place: one box per choice.
One box per choiceMultiply the number of choices for each place
From the lesson: Arrangements and selections
Know it
What is $n!$ , and what does it count?
Answer
$n \times (n - 1) \times \dots \times 2 \times 1$ : the ways to arrange $n$ different things in a row. So 5 people: $5! = 120$ . By agreement $0! = 1$ .
From the lesson: Arrangements and selections
Know it
What is ${}^nP_r$ ?
Answer
The arrangements of $r$ things from $n$ , where order matters: $\dfrac{n!}{(n - r)!}$ .
From the lesson: Arrangements and selections
Rule
What is ${}^nC_r$ ?
Answer
The selections of $r$ things from $n$ , where order doesn't matter: $\dfrac{n!}{r!\,(n - r)!}$ . Each selection gives $r!$ arrangements.
Selections and arrangementsEach selection of r things gives r! orders
From the lesson: Arrangements and selections
Know it
Permutation or combination: how do you decide?
Answer
Ask: if two of the chosen swapped places, would it be a different outcome? Yes: a permutation. No (a committee, a team): a combination.
From the lesson: Arrangements and selections
Know it
How many ways can the letters of STATISTICS be arranged?
Answer
Divide by the repeats: S three times, T three times, I twice, so $\dfrac{10!}{3!\,3!\,2!}$ .
From the lesson: Arrangements and selections
Rule
In how many ways can $n$ people sit round a table?
Answer
$(n - 1)!$ : fix one person's seat and arrange the rest. Six people: $5! = 120$ .
Round a tableTurning the table gives the same seating
From the lesson: Arrangements and selections
Which method?
JAMB 2010 · UTME · Q49
In how many ways can a committee of 2 women and 3 men be chosen from 6 men and 5 women?
How do the two choices combine?
Answer
Choose the women and the men separately, ${}^5C_2$ and ${}^6C_3$ , then multiply: "and" multiplies.
From the lesson: Arrangements and selections Try the question
Which method?
JAMB 2012 · UTME · Q48
In how many ways can the letters of the word TOTALITY be arranged?
What must you check first?
Answer
Repeated letters: TOTALITY has T three times. So it is $\dfrac{8!}{3!}$ , not $8!$ .
From the lesson: Arrangements and selections Try the question