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Five cards are drawn from a standard deck of 52 cards. What is the probability that there is an ace and a king or a queen among 5 cards.

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    What are your thoughts on the problem? Have you tried anything so far? – 2017-02-14
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    I think it should be (50 C 3 + 50 C 3)/(52 C 5) – 2017-02-14
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    @ShahzaibAli Put it in the post; not the comments. Also include *why* you think so. – 2017-02-14

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You want the probability of selecting at least one from four aces and at least one from eight king or queens when selecting any five from all fifty two cards.

It will be easier, I suggest, to calculate the complement using the principle of inclusion and exclusion (and deMorgan's rule).

$$\begin{align}\mathsf P\big(A\cap(K\cup Q)\big) ~&=~ 1 - \mathsf P\Big(\big(A\cap(K\cup Q)\big)^\complement\Big) \\[1ex] &\hspace{1.5ex}\vdots \\[1ex] &=~ 1 - \mathsf P\big(A^\complement\big) - \mathsf P\big((K\cup Q)^\complement\big) + \mathsf P\big((A\cup K\cup Q)^\complement\big) \end{align}$$

  • $\mathsf P\big(A^\complement\big)$ is the probability of selecting no aces and any five from the forty-eight other cards when selecting five from all fifty two.

  • $\mathsf P\big((K\cup Q)^\complement\big)$ is the probability of selecting no king or queen and any five from the forty-four other cards when selecting five from all fifty two.

  • $\mathsf P\big((A\cup K\cup Q)^\complement\big)$ is the probability of selecting no king, queen, or ace, and any five from the forty other cards when selecting five from all fifty two.