Final answer:
To find the number of selections with at least 3 jacks, we consider two cases: when exactly 3 jacks are chosen and when more than 3 jacks are chosen. We calculate the number of selections for each case using combination formulas. The total number of selections is the sum of the selections in both cases.
Step-by-step explanation:
To find the number of selections in which at least 3 jacks occur, we need to consider two cases: when exactly 3 jacks are chosen and when more than 3 jacks are chosen.
Case 1: Exactly 3 jacks are chosen.
We have 4 jacks in the deck, so we can choose 3 jacks in (4 choose 3) = 4 ways. The remaining 4 cards can be chosen in (48 choose 4) ways. Therefore, the number of selections in this case is 4 * (48 choose 4).
Case 2: More than 3 jacks are chosen.
We have 4 jacks in the deck, so we can choose more than 3 jacks in (4 choose 4) + (4 choose 5) + (4 choose 6) + (4 choose 7) + (4 choose 8) + (4 choose 9) + (4 choose 10) + (4 choose 11) + (4 choose 12) + (4 choose 13) ways. The remaining 3 cards can be chosen in (48 choose 3) ways. Therefore, the number of selections in this case is [(4 choose 4) + (4 choose 5) + (4 choose 6) + (4 choose 7) + (4 choose 8) + (4 choose 9) + (4 choose 10) + (4 choose 11) + (4 choose 12) + (4 choose 13)] * (48 choose 3).
To find the total number of selections with at least 3 jacks, we need to sum the number of selections from Case 1 and Case 2. Therefore, the total number of selections is 4 * (48 choose 4) + [(4 choose 4) + (4 choose 5) + (4 choose 6) + (4 choose 7) + (4 choose 8) + (4 choose 9) + (4 choose 10) + (4 choose 11) + (4 choose 12) + (4 choose 13)] * (48 choose 3).