If the total number of one-one functions from A = {2, 3, 4} to {1, 5, 6, 7} are 24 and the total number of many-one functions are (5k + 15), then the value of k is
Question
If the total number of one-one functions from A = {2, 3, 4} to {1, 5, 6, 7} are 24 and the total number of many-one functions are (5k + 15), then the value of k is
Solution
The total number of functions from a set A (with m elements) to a set B (with n elements) is given by n^m.
In this case, m = 3 (as set A has 3 elements) and n = 4 (as the second set has 4 elements). So, the total number of functions is 4^3 = 64.
The problem states that the total number of one-one (or injective) functions is 24.
The total number of many-one (or non-injective) functions is the total number of functions minus the number of one-one functions.
So, the total number of many-one functions is 64 - 24 = 40.
The problem states that this number is equal to 5k + 15. So, we can set up the equation 5k + 15 = 40.
Solving for k, we subtract 15 from both sides to get 5k = 25, and then divide both sides by 5 to get k = 5.
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