Let A_k be a set containing k elements for k 3, 4, 5 , and B be a set containing 6 elements. Let C be a specific subset of B containing exactly 2 elements. If the total number of functions from A_k to B (across all three values of k ) such that the image of the function contains at least one element of C is 8 , then the value of is

Step-by-step solution

For a fixed k , the total number of functions from A_k to B is given by |B|^ |A_k| = 6^k . The number of functions whose image does not contain any element of C is equivalent to the number of functions mapping entirely into B C . Since |B C| = 6 - 2 = 4 , this number is 4^k . Thus, the number of valid functions for a given k is 6^k - 4^k . Summing this over k = 3, 4, 5 , we get the total number of valid functions: _ k=3 ⁵ (6^k - 4^k) = (6^3 - 4^3) + (6^4 - 4^4) + (6^5 - 4^5) Calculating each term: For k=3 : 216 - 6