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Set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

A is a subset of X and has exactly 4 elements.
Find the number of subsets A that satisfy these conditions:
a) A already contains 3 and 5.
b) The largest element in A is 8.
c) The smallest element in A is 2 OR 3.

User Csviri
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1 Answer

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Final answer:

The number of subsets A of set X that contain exactly 4 elements, include 3 and 5, have a largest element of 8, and have a smallest element of 2 (since 3 is already included), is calculated by selecting 2 elements from the remaining options {1, 2, 4, 6, 7, 8}, which is C(5, 2).

Step-by-step explanation:

We are given set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and we need to find the number of subsets A that satisfy certain conditions, with A being a subset of X that has exactly 4 elements.

  • Condition a): Subset A already contains 3 and 5. Since two of the four elements are already fixed, we only need to choose 2 more elements from the remaining elements of set X that are not 3 or 5.
  • Condition b): The largest element in A is 8. Thus, our options for selecting the remaining two elements are limited to {1, 2, 4, 6, 7, 8}.
  • Condition c): The smallest element in A is 2 or 3. However, since 3 is already included in A, we only consider subsets where the smallest element is 2, which leaves us with {1, 2, 4, 6, 7, 8} - {3, 5} = {1, 2, 4, 6, 7, 8} for our selection.

To find the number of possible subsets satisfying all conditions, we need to select 2 distinct elements out of 5 options (since elements 3 and 5 are fixed), which corresponds to the number of combinations of 5 items taken 2 at a time (using the combination formula C(n, k) = n! / (k!(n-k)!) where n is the total number and k is the number to choose). The calculation is C(5, 2).

User Augre
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