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# Counting question

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How many subsets of the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} contain the number 5 and have 3 elements?

Dec 12, 2021

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There are: 10 C 3 ==120 subset of 3 digits each as follows:

(1, 2, 3) , (1, 2, 4) , (1, 2, 5) , (1, 2, 6) , (1, 2, 7) , (1, 2, 8) , (1, 2, 9) , (1, 2, 10) , (1, 3, 4) , (1, 3, 5) , (1, 3, 6) , (1, 3, 7) , (1, 3, 8) , (1, 3, 9) , (1, 3, 10) , (1, 4, 5) , (1, 4, 6) , (1, 4, 7) , (1, 4, 8) , (1, 4, 9) , (1, 4, 10) , (1, 5, 6) , (1, 5, 7) , (1, 5, 8) , (1, 5, 9) , (1, 5, 10) , (1, 6, 7) , (1, 6, 8) , (1, 6, 9) , (1, 6, 10) , (1, 7, 8) , (1, 7, 9) , (1, 7, 10) , (1, 8, 9) , (1, 8, 10) , (1, 9, 10) , (2, 3, 4) , (2, 3, 5) , (2, 3, 6) , (2, 3, 7) , (2, 3, 8) , (2, 3, 9) , (2, 3, 10) , (2, 4, 5) , (2, 4, 6) , (2, 4, 7) , (2, 4, 8) , (2, 4, 9) , (2, 4, 10) , (2, 5, 6) , (2, 5, 7) , (2, 5, 8) , (2, 5, 9) , (2, 5, 10) , (2, 6, 7) , (2, 6, 8) , (2, 6, 9) , (2, 6, 10) , (2, 7, 8) , (2, 7, 9) , (2, 7, 10) , (2, 8, 9) , (2, 8, 10) , (2, 9, 10) , (3, 4, 5) , (3, 4, 6) , (3, 4, 7) , (3, 4, 8) , (3, 4, 9) , (3, 4, 10) , (3, 5, 6) , (3, 5, 7) , (3, 5, 8) , (3, 5, 9) , (3, 5, 10) , (3, 6, 7) , (3, 6, 8) , (3, 6, 9) , (3, 6, 10) , (3, 7, 8) , (3, 7, 9) , (3, 7, 10) , (3, 8, 9) , (3, 8, 10) , (3, 9, 10) , (4, 5, 6) , (4, 5, 7) , (4, 5, 8) , (4, 5, 9) , (4, 5, 10) , (4, 6, 7) , (4, 6, 8) , (4, 6, 9) , (4, 6, 10) , (4, 7, 8) , (4, 7, 9) , (4, 7, 10) , (4, 8, 9) , (4, 8, 10) , (4, 9, 10) , (5, 6, 7) , (5, 6, 8) , (5, 6, 9) , (5, 6, 10) , (5, 7, 8) , (5, 7, 9) , (5, 7, 10) , (5, 8, 9) , (5, 8, 10) , (5, 9, 10) , (6, 7, 8) , (6, 7, 9) , (6, 7, 10) , (6, 8, 9) , (6, 8, 10) , (6, 9, 10) , (7, 8, 9) , (7, 8, 10) , (7, 9, 10) , (8, 9, 10)>>Total==120 such subsets.

There are: 2 * 3! + 4!==36 subsets with a "5" in them!

Dec 13, 2021