The number of tens in a standard 52-card deck is 4. Since we are selecting 4 tens from 4 available, there is 1 way to do it.
Next, we are choosing one card from the 52-standard card deck without choosing any tens. Since there are 4 tens in a deck and we don't want to choose any of them, we subtract 4 from the total number of cards (52), which leaves us with 48 cards. We can choose one card from these 48 in 48 ways.
To calculate the number of ways to choose a five-card hand containing exactly four tens, we have already determined that there is 1 way to choose 4 tens. We then have to choose one more card out of the remaining cards not being tens. Since there are 48 cards available (from previous calculation), the number of ways to get the additional card is simply 48. Multiplying these numbers together gives us 48 ways.
In order to find out the number of five-card hands that contain a four-of-a-kind, we first need to determine the number of ways to pick a set of four identical rank cards (other than tens as they have already been accounted for in the initial part of the question). The number of cards left in deck excluding the tens is 48. From these 48, we can choose 4 identical rank cards in 9339840 ways (combination formula). Then, we pick one more card from the rest of the deck (which is 52 minus the 4 identical rank cards) in 48 ways. Multiplying those two values, we end up with 9339840.
So to summarize, there is 1 way to choose four tens from a standard 52-card deck, 48 ways to choose one card from a standard 52-card deck without choosing any tens, 48 ways for a five-card hand to contain exactly four tens, and 9339840 ways for a five-card hand to contain a four-of-a-kind.