[Math] How many ways to select 5 cards with at least one king.

Question

I am given a deck of $52$ cards in which I have to select $5$ card which has at least one king.
So I selected one king out of $4$ and then remaining from deck as
$$^4C_1. ^{51}C_4$$ which however gives wrong answer but subtracting total combination from combination with no gives right answer
$$^{52}C_5- ^{48}C_5$$ So, why is first approach wrong?

Answer 1

You are "duplicating combinations", because the same king that you choose out of $4$ kings in one combination, can be chosen out of $51$ cards in another combination.

Instead, calculate the total number of combinations, and then subtract the number of combinations with no kings at all:

$$\binom{52}{5}-\binom{52-4}{5}$$