Permutations with Repetition — Advanced

1) Find the number of distinct arrangements of “STATISTICS”.

50,400

52,000

48,600

54,000

“STATISTICS” has 10 letters: S=3, T=3, I=2 → 10!/(3!·3!·2!) = 3,628,800/72 = 50,400.

Explanation

“STATISTICS” has 10 letters: S=3, T=3, I=2 → 10!/(3!·3!·2!) = 3,628,800/72 = 50,400.

2) Find the number of distinct arrangements of the digits in 122333.

60

72

54

80

Digits 1,2,2,3,3,3 → 6!/(1!·2!·3!) = 720/12 = 60.

Explanation

Digits 1,2,2,3,3,3 → 6!/(1!·2!·3!) = 720/12 = 60.

3) Find the number of distinct arrangements of the letters in “BOOKKEEPERS”.

831,600

840,000

820,000

850,000

“BOOKKEEPERS” has 11 letters: O=2, K=2, E=3 → 11!/(2!·2!·3!) = 39,916,800/48 = 831,600.

Explanation

“BOOKKEEPERS” has 11 letters: O=2, K=2, E=3 → 11!/(2!·2!·3!) = 39,916,800/48 = 831,600.

4) Find the number of distinct 8-letter arrangements in “BALLOONS”.

10,080

10,500

9,600

11,000

“BALLOONS” has 8 letters with L=2, O=2 → 8!/(2!·2!) = 40,320/4 = 10,080.

Explanation

“BALLOONS” has 8 letters with L=2, O=2 → 8!/(2!·2!) = 40,320/4 = 10,080.

5) Find the number of distinct arrangements of “ARRANGEMENT”.

831,600

820,000

840,000

850,000

“ARRANGEMENT” has 11 letters: A=2, R=2, N=2 → 11!/(2!·2!·2!) = 39,916,800/48 = 831,600.

Explanation

“ARRANGEMENT” has 11 letters: A=2, R=2, N=2 → 11!/(2!·2!·2!) = 39,916,800/48 = 831,600.

6) Find the number of distinct arrangements of the letters in “SUCCESSFUL”.

151,200

155,000

148,000

160,000

“SUCCESSFUL” has 10 letters: S=3, C=2, U=2 → 10!/(3!·2!·2!) = 3,628,800/24 = 151,200.

Explanation

“SUCCESSFUL” has 10 letters: S=3, C=2, U=2 → 10!/(3!·2!·2!) = 3,628,800/24 = 151,200.

7) Find the number of distinct arrangements of “SCHOOLBUS”.

181,440

185,000

178,000

190,000

“SCHOOLBUS” has 9 letters with O repeated twice → 9!/2! = 362,880/2 = 181,440.

Explanation

“SCHOOLBUS” has 9 letters with O repeated twice → 9!/2! = 362,880/2 = 181,440.

8) Find the number of distinct arrangements of “REPETITION”.

453,600

460,000

440,000

470,000

“REPETITION” has 10 letters: T=2, I=2, O=2 → 10!/(2!·2!·2!) = 3,628,800/8 = 453,600.

Explanation

“REPETITION” has 10 letters: T=2, I=2, O=2 → 10!/(2!·2!·2!) = 3,628,800/8 = 453,600.

9) Find the number of distinct arrangements of “TENNESSEE”.

37,800

38,000

36,000

40,000

“TENNESSEE” has 9 letters: E=4, N=2, S=2 → 9!/(4!·2!·2!) = 362,880/9,600 = 37,800.

Explanation

“TENNESSEE” has 9 letters: E=4, N=2, S=2 → 9!/(4!·2!·2!) = 362,880/9,600 = 37,800.

10) Find the number of arrangements of the digits in 1000000.

7

6

8

9

7 digits: one '1' and six zeros → 7!/6! = 7.

Explanation

7 digits: one '1' and six zeros → 7!/6! = 7.