I was doing a seemingly trivial question, and I though it was a simple application of the counting theorem but it turns out it doesn't work. Here's the questionFrom a deck of 52 cards, how many ways are there to arrange a hand of 5 cards such that all 4 kings are in the hand (order doesn't matter) (the last card can be any non-king)Now here's my thought process as an application of the counting principle: 4 × 3 × 2 × 1 × 48 5 ! As we have 4! ways of placing the kings and then the last card can be from 48 other cards. Then we divide by 5! to remove the order. Unfortunately, this produces a non-integer so I was very sad indeed. However, it logically seems like it should work as it follows what I think is valid logic. Could someone explain how to get the correct answer (48) and also more importantly, why my logic was incorrect?

Question

I was doing a seemingly trivial question, and I though it was a simple application of the counting theorem but it turns out it doesn&#039;t work. Here&#039;s the questionFrom a deck of 52 cards, how many ways are there to arrange a hand of 5 cards such that all 4 kings are in the hand (order doesn&#039;t matter) (the last card can be any non-king)Now here&#039;s my thought process as an application of the counting principle:            4      ×      3      ×      2      ×      1      ×      48              5      !      As we have 4! ways of placing the kings and then the last card can be from 48 other cards. Then we divide by 5! to remove the order. Unfortunately, this produces a non-integer so I was very sad indeed. However, it logically seems like it should work as it follows what I think is valid logic. Could someone explain how to get the correct answer (48) and also more importantly, why my logic was incorrect?

Jazmyn Bean · Accepted Answer

When you write  4  ×  3  ×  2  ×  1  ×  48you haven&#039;t chosen yet exactly where the non-king is in the sequence.Yet when you divide by 5! you pretend that you have done so. That&#039;s where your argument fails.

vroos5p · Accepted Answer

Nice

From a deck of 52 cards, how many ways are there to arrange a hand of 5 cards such that all 4 kings are in the hand (order doesn't matter) (the last card can be any non-king)

Open question

Answer & Explanation

New Questions in High school probability