Answer:
(a) 129
(b) 5
Step-by-step explanation:
(a) In a set with [tex]n[/tex] elements, the number of possible distinct subsets is [tex]2^n[/tex].
For a 6-element set, this number is [tex]2^6 = 64[/tex].
The minimum for two identical subsets will be 1 + 64 = 65, since 64 already contains all possible subsets. Any extra subset will be identical to one of them.
For three identical subsets, all subsets must have been repeated twice. Any extra is then repeated thrice. Hence, we have 64 × 2 + 1 = 129
(b) With 257 subsets, we first determine how many times each distinct subset could have been repeated by dividing 257 ÷ 64 = 4 with a remainder of 1. This means possibly, all subsets have been repeated 4 times. The extra 1 makes it a fifth time for one of them. So, the minimum number of identical subsets is 5.
The number of sets that must be printed to be sure of having at least 3 identical subsets on the list is 129.
The sets that must be printed to be sure of having at least 3 identical subsets on the list will be calculated thus:
= (26 × 2) + 1
= 128 + 1
= 129
The number of identical subsets that are printed if there are 257 subsets on the list will be:
= (257/64) + 1
= 4 + 1
= 5
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