The 6-shop

Probability Level 3

In this shop they sell exactly 6 different articles, but there are far more than 6 exemplars of each article. Somebody buys 4 articles, eventual repetitions included. How many outcomes are possible?


The answer is 126.

Kees Vugs by Kees Vugs , 87, Netherlands

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2 solutions

Jesse Li
Nov 2, 2018

We can solve this problem with casework.

Case 1: No matches

Case 2: 2 matching

Case 3: 3 matching

Case 4: All matching

Case 5: 2 matching+2 matching

Case 1: There are ( 6 4 ) 6 \choose 4 = 15 ways to choose 4 articles

Case 2: There are 6 × ( 5 2 ) 6 \times {5 \choose 2} = 60 ways to choose 4 articles

Case 3: There are 6 × 5 6 \times 5 = 30 ways to choose 4 articles

Case 4: There are 6 ways to choose 4 articles

Case 5: There are ( 6 2 ) 6 \choose 2 = 15 ways to choose 4 articles

15 + 60 + 30 + 6 + 15 15 + 60 + 30 + 6 + 15 = 126 \boxed {126}

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Kees Vugs
Nov 2, 2018

Call x(i) the number of bought articles of kind i. Then we have to find the number of solutions (x(1), x(2), . . ., x(6)) of the linear equation x(1) + x(2) + . . . + x(6) = 4 with x(i) a natural number (eventually 0). This number is equal to the number of ways upon which in a row of (4 + 6 - 1) objects one can indicate (6 - 1) objects (the "+" signs). This number is ( 4 + 6 − 1 ) ! ( 6 − 1 ) ! 4 ! \frac{(4 + 6 - 1)!}{(6 - 1)! 4!} = 126

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