I tried the Fundamental Counting principle approach: There are 26 black cards to chose from for the first card. That leaves 26 × 25 × 24 × 23 left for the other four cards. So 26 × 26 × 25 × 24 × 23 5 ! (because we can arrange 5 cards in 5 ! ways). I get 77740. Apparently the answer is: 454480.I'd appreciate help clearing up my misunderstanding. I'd also like to see the combinations approach. Thanks!

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I tried the Fundamental Counting principle approach: There are 26 black cards to chose from for the first card. That leaves   26  ×  25  ×  24  ×  23 left for the other four cards. So             26      ×      26      ×      25      ×      24      ×      23              5      !       (because we can arrange 5 cards in   5  ! ways). I get 77740. Apparently the answer is: 454480.I&#039;d appreciate help clearing up my misunderstanding. I&#039;d also like to see the combinations approach. Thanks!

Kyan Mcdonald · Accepted Answer

You missed two things, I believe:1) You want at most one black card. That means you still have to consider the possibility of all cards being black.2) When counting the cases in which 4 cards are black and one is black like you did, you don&#039;t have 5! ways of rearranging, you have less. This is because, as you said, there are 26 black cards to choose from for the first card. You&#039;re never considering the case of choosing a black card second or third, so you shouldn&#039;t consider rearrangements in which you swap the black card. Only rearrangements of the 4 black cards. Does this make sense?And, in fact, combining these two points:            26      ×      25      ×      24      ×      23      ×      22              5      !        +            26      ×      26      ×      25      ×      24      ×      23              4      !        =  454480

clovnerie0q · Accepted Answer

Using combinatorics:We have                     (                    26      4                      )                                (                    26      1                      )              +                    (                    26      5                      )                                (                    26      0                      )              =  454480The first bit is choosing 4 black cards and 1 black card. Then the other scenario is only choosing 5 black cards! Hope this helps!

There are 26 black cards to chose from for the first card. That leaves 26252423 left for the other four cards. So 26262524*23/5! (because we can arrange 5 cards in 5! ways). I get 77740. Apparently the answer is: 454480. I'd appreciate help clearing up my misunderstanding.

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