3D - 2

Permutation or Combination?

in the Magnum 4D there are 10 000 numbers as follows: 0000,0001,0002,...,9998,9999. how many among these numbers a. have 2 pairs of similar digits? b. have a digit occurring 3 times and a different digit occurring once? c. have a repeating digit and 2 other digits different? d. have all 4 digits different? e. have all digits the same?

Public Comments

1. a mix of permutations and combinations basically the solution follows a uniform pattern [from 10 #s, choose k distinct #s as specified] x [ permutate 4 digits taking care of repetitions, if any] a. 10C2 • 4! / (2! • 2!) = 270 b. 10C2 • 4! / 3! = 180 c. 10C3 • 4! / 2! = 1440 d. 10C4 • 4! = 5040 e. 10C1 = 10
2. a. 10 numbers choosing 2 {because 2 similar number} ≡ ¹ºC₂ arranging 4 numbers with 2 similar number ≡ 4! / (2! x 2!) ¹ºC₂ x 4! / (2! x 2!) = 270 b. 10 numbers choosing 2 {because 2 different numbers used only} ≡ ¹ºC₂ arranging 4 numbers with a number repeated 3 times ≡ 4! / (3! x 1!) ¹ºC₂ x 4! / (3! x 1!) = 180 c. 10 numbers choosing 3 {because 3 different numbers used only} ≡ ¹ºC₃ arranging 4 numbers with a number repeated twice ≡ 4! / (2! x 1! x 1!) ¹ºC₃ x 4! / (2! x 1! x 1!) = 1440 d. arranging 4 digits from 10 digits without repetition ≡ ¹ºP₄ ¹ºP₄ = 5040 e. choosing 4 numbers from 10 numbers without repetition ≡ ¹ºC₁ ¹ºC₁ = 10