Permutations & Combinations

Solution

8P3 = 8 x 7 x 6 = 336 Answer: 336

Solution

10C4 = (10 x 9 x 8 x 7)/(4 x 3 x 2 x 1) = 5040/24 = 210 Answer: 210

Solution

12C10 = 12C2 (since nCr = nC(n-r)) = (12 x 11)/2 = 66 Answer: 66

Solution

LOGIC has 5 distinct letters. Arrangements = 5! = 120 Answer: 120

Solution

Arrangements of 6 distinct books = 6! = 720 Answer: 720

Solution

Order does not matter, so use combinations. 9C3 = (9 x 8 x 7)/(3 x 2 x 1) = 84 Answer: 84

Solution

Handshakes = 12C2 = (12 x 11)/2 = 66 Answer: 66

Solution

Each of the 3 places can be filled in 5 ways. Total = 5 x 5 x 5 = 125 Answer: 125

Solution

Diagonals = nC2 - n = 8C2 - 8 = 28 - 8 = 20 Answer: 20

Solution

The posts are different, so order matters. 8P2 = 8 x 7 = 56 Answer: 56

Solution

By the multiplication rule: 4 x 3 x 2 = 24 Answer: 24

Solution

If nCa = nCb with a not equal to b, then a + b = n. r + 4 = 12, so r = 8 Answer: 8

Solution

Matches = 8C2 = (8 x 7)/2 = 28 Answer: 28

Solution

nP2 = n(n-1) = 42 = 7 x 6 So n = 7 Answer: 7

Solution

Each of the 4 places has 10 choices. Total = 10^4 = 10,000 Answer: 10,000

Solution

Letters: B x 1, A x 3, N x 2; total 6 letters. Arrangements = 6!/(3! x 2!) = 720/12 = 60 Answer: 60

Solution

Letters: S x 3, U x 1, C x 2, E x 1; total 7 letters. Arrangements = 7!/(3! x 2!) = 5040/12 = 420 Answer: 420

Solution

Vowels: A, I, U (3 vowels). Treat AIU as one unit. Units: J, P, R and (AIU) = 4 units, arranged in 4! = 24 ways. Vowels inside the unit: 3! = 6 ways. Total = 24 x 6 = 144 Answer: 144

Solution

Total arrangements = 5! = 120 Vowels E, I together: units D, L, H, (EI) = 4! x 2! = 48 Not together = 120 - 48 = 72 Answer: 72

Solution

Cases: 2 women + 2 men: 4C2 x 5C2 = 6 x 10 = 60 3 women + 1 man: 4C3 x 5C1 = 4 x 5 = 20 4 women: 4C4 = 1 Total = 60 + 20 + 1 = 81 Answer: 81

Solution

Ravi is fixed, so choose the remaining 4 from 7 players. 7C4 = 35 Answer: 35

Solution

Thousands place: not 0, so 9 choices. Hundreds: 9 choices (0 allowed, one digit already used). Tens: 8 choices. Units: 7 choices. Total = 9 x 9 x 8 x 7 = 4,536 Answer: 4,536

Solution

Units digit must be 2 or 4: 2 choices. Tens digit: 4 remaining choices. Hundreds digit: 3 remaining choices. Total = 2 x 4 x 3 = 24 Answer: 24

Solution

Circular arrangements of n people = (n-1)! (7-1)! = 6! = 720 Answer: 720

Solution

Treat the two persons as one unit: 7 units in a circle = (7-1)! = 720 ways. The two can swap inside: 2! = 2. Total = 720 x 2 = 1,440 Answer: 1,440

Solution

Arrange the 5 boys first: 5! = 120 ways. This creates 6 gaps (including both ends). Choose 3 gaps for the girls: 6C3 = 20. Arrange the girls in them: 3! = 6. Total = 120 x 20 x 6 = 14,400 Answer: 14,400

Solution

A rectangle needs 2 of the 9 horizontal lines and 2 of the 9 vertical lines. Total = 9C2 x 9C2 = 36 x 36 = 1,296 Answer: 1,296

Solution

Total triples = 10C3 = 120 Triples from the 4 collinear points (no triangle) = 4C3 = 4 Triangles = 120 - 4 = 116 Answer: 116

Solution

Alphabetical order of letters: D, E, H, I, L. Words starting with D or E: 2 x 4! = 48 Starting with H: second letter D or E gives 2 x 3! = 12 words before HI... Starting with HI: third letter D gives HID.., and HIDEL is the first of these (E before L). Rank = 48 + 12 + 1 = 61 Answer: 61

Solution

Place 2 dividers in the 9 gaps between the 10 sweets. Ways = 9C2 = 36 Answer: 36

Solution

Each path is an arrangement of 6 R's and 4 U's. Paths = 10!/(6! 4!) = 10C4 = 210 Answer: 210

Solution

Total ways = 9C3 = 84 All fiction = 4C3 = 4; all non-fiction = 5C3 = 10 Required = 84 - 4 - 10 = 70 Answer: 70

Solution

Choose 4 for one team: 8C4 = 70. The two teams are unlabelled, so divide by 2: 70/2 = 35 Answer: 35

Solution

Total = 6! = 720 Together: treat as one unit: 5! x 2! = 240 Not together = 720 - 240 = 480 Answer: 480

Solution

Two players are fixed, so choose the other 9 from the remaining 13. 13C9 = 13C4 = (13 x 12 x 11 x 10)/24 = 715 Answer: 715

Solution

A number is divisible by 4 if its last two digits form a multiple of 4. Endings with distinct digits from 0-4: 04, 12, 20, 24, 32, 40. Endings containing 0 (04, 20, 40): the other 3 digits fill the first 3 places freely: 3! = 6 each -> 18 Endings without 0 (12, 24, 32): 0 is among the remaining 3 digits and cannot be first: 3! - 2! = 4 each -> 12 Total = 18 + 12 = 30 Answer: 30

Solution

Each digit appears in each place 3! = 6 times. Sum of digits = 1 + 2 + 3 + 4 = 10 Sum for each place = 6 x 10 = 60 Total = 60 x (1000 + 100 + 10 + 1) = 60 x 1111 = 66,660 Answer: 66,660

Solution

Derangements: D(n) = n! x (1/2! - 1/3! + 1/4! - ... ) D(5) = 120 x (1/2 - 1/6 + 1/24 - 1/120) = 60 - 20 + 5 - 1 = 44 Answer: 44

Solution

Letters: M x2, A x2, T x2, H, E, I, C, S: 11 letters. Arrangements = 11!/(2! 2! 2!) = 39916800/8 = 49,89,600 Answer: 49,89,600

Solution

Seat the 5 men first around the table: (5-1)! = 24 ways. The 5 women take the 5 gaps between men: 5! = 120 ways. Total = 24 x 120 = 2,880 Answer: 2,880

Solution

Squares of side k: (9-k)^2 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 Answer: 204

Solution

The thousands digit must be 3, 4 or 5: 3 choices. (A number starting with 3 and having distinct digits is always above 3000.) The other 3 places use the remaining 5 digits: 5 x 4 x 3 = 60 Total = 3 x 60 = 180 Answer: 180

Solution

Person 1: 6C2 = 15 ways; Person 2: 4C2 = 6 ways; Person 3: 2C2 = 1 way. Total = 15 x 6 x 1 = 90 Answer: 90

Solution

Total arrangements = 5!/(2! 2!) = 30 A's together: units (AA), B, B, C = 4!/2! = 12; B's together: 12; both together: (AA), (BB), C = 3! = 6 At least one pair together = 12 + 12 - 6 = 18 No identical letters adjacent = 30 - 18 = 12 Answer: 12

Solution

Vowels E, A, E occupy positions 1, 3, 5: 3!/2! = 3 ways. Consonants L, D, R occupy positions 2, 4, 6: 3! = 6 ways. Total = 3 x 6 = 18 Answer: 18