# A bag contains club,heart,spade,star, and circle. How many different sets of 4 of these can be formed?

justaguide | College Teacher | (Level 2) Distinguished Educator

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The bag contains clubs, hearts, spades, stars, and circles. Sets of 4 items from the bag are being formed. There are no restrictions on the number of times an item of any type can be included in the set. Each of the 5 types of items can be included any number of times in the set that is being created.

The number of combinations of r items chosen from a set containing n items when repetition is allowed is given by `((n + r - 1)!)/(r!*(n-1)!)`

Substituting the values in the problem, n = 5 and r = 4. This gives the total number of sets that can be created as `((5+4-1)!)/(4!*(5-1)!)` = `(8!)/(4!*4!)` = 70

The number of sets that can be formed is 70.

degeneratecircle | High School Teacher | (Level 2) Associate Educator

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As worded, the answer is five. Since the answer is supposed to be 70 (the answer was originally given by the question asker), this is a very poorly worded problem. If there are no restrictions on the number of each type chosen, then clearly there are infinitely many ways to pick them:

1 club, 1 heart, 1 spade, 1 star

2 clubs, 1 heart, 1 spade, 1 star

3 clubs, 1 heart, 1 spade, 1 star, and so on.

Are you sure this is exactly how the question is worded?

Oh, "4 of these" means four cards, and not four suits. Now the answer of 70 makes sense. It's still ambiguously worded, since three of us misinterpreted it.

degeneratecircle | High School Teacher | (Level 2) Associate Educator

Posted on

As worded, the answer is five. Since the answer is supposed to be 70 (the answer was originally given by the question asker), this is a very poorly worded problem. If there are no restrictions on the number of each type chosen, then clearly there are infinitely many ways to pick them:

1 club, 1 heart, 1 spade, 1 star

2 clubs, 1 heart, 1 spade, 1 star

3 clubs, 1 heart, 1 spade, 1 star, and so on.

Are you sure this is exactly how the question is worded?