In an ordered (A – K, ♠♢♣♡) 52 card deck, using JUMP SEARCH, looking for the queen of hearts,

a. What is the optimum jump size k?
b. How many probes (m) will it take with the optimum jump size?
c. What is the worst case number of comparisons using k and m?

To solve this problem using the jump search algorithm, let's go through each step:

a. What is the optimum jump size k?

The optimum jump size can be calculated using the formula √n, where n is the size of the array or the number of elements. In this case, the number of cards in the deck is 52. So the optimum jump size (k) will be √52, which is approximately 7.21; we can round it down to 7 for simplicity.

b. How many probes (m) will it take with the optimum jump size?

To find the number of probes (m), divide the size of the array (n) by the jump size (k). In this case, the size of the array is 52, and the jump size is 7. So the number of probes (m) will be 52 / 7, which is approximately 7.42; we can round it up to 8.

c. What is the worst case number of comparisons using k and m?

The worst-case scenario occurs when the element being searched is at the last index of the array or cannot be found at all. In this case, the worst-case number of comparisons would be (m + k - 1) if the element is found, or (m) if the element is not found.

So in our case, with the optimal jump size (k = 7) and number of probes (m = 8), the worst-case number of comparisons would be (8 + 7 - 1) = 14, if the queen of hearts is found, and (8) if it is not found.

Therefore:
a. The optimum jump size (k) is 7.
b. The number of probes (m) with the optimum jump size (k) is 8.
c. The worst-case number of comparisons using the optimum jump size (k) and number of probes (m) is 14 if the queen of hearts is found, and 8 if it is not found.