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Especially Lime
Especially Lime
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There are two issues here. The first is that the standard inclusion-exclusion formula assumes you're trying to subtractyou start by subtracting events where at least one thing is missing, so that everything where two are missing is counted twice, three missing counted three times, etc, and that is what means you only have to add or subtract once, and which you do alternates.

Here you are trying to countstart by subtracting things where two things are missing. This means that you counthave subtracted everything with three missing three times (from three suits you can choose a pair in three ways), and so you need to add on twice the number of ways to have three suits missing. (If it was a possible situation, you'd then need to add, rather than subtract, three times the number of ways to have four missing: you would have so far subtracted these configurations six times and added them foureight times.)

The second problem is that you haven't accounted for all the situations where all four suits are present. So after making the change from the previous paragraph, what you have calculated is the number of ways to have at least three suits among the chosen cards.

There are two issues here. The first is that the standard inclusion-exclusion formula assumes you're trying to subtract events where at least one thing is missing, so that everything where two are missing is counted twice, three missing counted three times, etc, and that is what means you only have to add or subtract once, and which you do alternates.

Here you are trying to count things where two things are missing. This means that you count everything with three missing three times (from three suits you can choose a pair in three ways), and so you need to add on twice the number of ways to have three suits missing. (If it was a possible situation, you'd then need to add, rather than subtract, the number of ways to have four missing: you would have so far subtracted these configurations six times and added them four times.)

The second problem is that you haven't accounted for all the situations where all four suits are present. So after making the change from the previous paragraph, what you have calculated is the number of ways to have at least three suits among the chosen cards.

There are two issues here. The first is that the standard inclusion-exclusion formula assumes you start by subtracting events where at least one thing is missing, so that everything where two are missing is counted twice, three missing counted three times, etc, and that is what means you only have to add or subtract once, and which you do alternates.

Here you start by subtracting things where two are missing. This means that you have subtracted everything with three missing three times (from three suits you can choose a pair in three ways), and so you need to add on twice the number of ways to have three suits missing. (If it was a possible situation, you'd then need to subtract three times the number of ways to have four missing: you would have so far subtracted these configurations six times and added them eight times.)

The second problem is that you haven't accounted for all the situations where all four suits are present. So after making the change from the previous paragraph, what you have calculated is the number of ways to have at least three suits among the chosen cards.

Source Link
Especially Lime
Especially Lime
  • 41.8k
  • 9
  • 57
  • 86

There are two issues here. The first is that the standard inclusion-exclusion formula assumes you're trying to subtract events where at least one thing is missing, so that everything where two are missing is counted twice, three missing counted three times, etc, and that is what means you only have to add or subtract once, and which you do alternates.

Here you are trying to count things where two things are missing. This means that you count everything with three missing three times (from three suits you can choose a pair in three ways), and so you need to add on twice the number of ways to have three suits missing. (If it was a possible situation, you'd then need to add, rather than subtract, the number of ways to have four missing: you would have so far subtracted these configurations six times and added them four times.)

The second problem is that you haven't accounted for all the situations where all four suits are present. So after making the change from the previous paragraph, what you have calculated is the number of ways to have at least three suits among the chosen cards.