Exercises: Combinations

30

720

2

24

Combinations use smaller numbers in the formula

Combinations ignore the order of the selected items, so each group is counted once instead of multiple times

Combinations only apply when fewer than half the items are selected

Permutations add items while combinations subtract them

Yes — assigning Ana as president and Ben as VP is different from assigning Ben as president and Ana as VP

No — the same two people are chosen either way

Only if the club has more than 10 members

Only if the president and VP have different responsibilities

56

336

24

168

21

42

2520

35

10

20

25

60

$C(6, 3) = \frac{6!}{3! \cdot 3!}$

$P(6, 3) = \frac{6!}{3!}$

$6^3$

$6 \cdot 3!$

The student computed $P(12, 4)$ instead of $C(12, 4)$ — course selection has no ranking, so divide by $r! = 24$ to get 495

The student swapped $n$ and $r$ — it should be $C(4, 12)$

The student should have multiplied by $r!$ instead of dividing

The student used the wrong value for
$\frac{12!}{8!}$

Jamal used the permutation formula instead of the combination formula — choosing books has no ranking, so he should divide by $r! = 6$

Jamal computed $7 \times 6 \times 5$ incorrectly

Jamal should have used $n = 3$ and $r = 7$

Jamal forgot to subtract 1 from 7 before computing

Dina swapped $n$ and $r$ — the total pool is $n = 9$ and the selection size is $r = 2$, giving $C(9, 2) = 36$

Dina should have used the permutation formula instead

The problem really is impossible because 2 is too small

Dina forgot to add 1 to each number

90

210

5040

30