Respuesta :
Answer: The correct option is (D) [tex]^{12}C_1\times ^{51}P_{51}.[/tex]
Step-by-step explanation: Given that there are 12 face cards in a standard deck of 52 cards.
We are to find the number of ways in which we can arrange a standard deck of 52 cards such that the first card is a face card.
Now, since there are 12 face cards, and the first card may be any one of theses 12 cards, so the number of ways in which the first card of 52 cards can be arranged is given by
[tex]n_1=^{12}C_1.[/tex]
Also, there are 51 cards left after the first card is arranged.
Now, out of these 51 cards, we need to choose all the 51 cards one by one and this is a situation of permutation.
So, the number of ways in which these 51 cards can be arranged is
[tex]n_2=^{51}P_{51}.[/tex]
Therefore, the total number of ways in which we can arrange a standard deck of 52 cards such that the first card is a face card will be
[tex]N=n_1\times n_2=^{12}C_1\times ^{51}P_{51}.[/tex]
Thus, (D) is the correct option.
