How many permutations of {1, 2, 3, 4, 5, 6} exist such that: (a) No even integer is in its natural position. (b) Exactly three integers are in their natural position. (c) 1 and 6 are in their natural positions.

(a) 6 permutations

(b) 6 permutations

(c) 24 permutations

Step-by-step explanation:

Given the set {1, 2, 3, 4, 5, 6}

We need to check how many permutations exist such that:

(a) No even integer is in its natural position.

All we need to do is count the number of elements, n, between which we want to make permutations, and take factorial of n to know the number of permutations.

The even integers are 2, 4, and 6, and we want to ensure that they are in their natural positions. So, the permutations are between 3 elements. For n = 3, factorial of 3 is 3! = 3×2×1 = 6.

To back this up, see the permutations.

{1, 2, 3, 4, 5, 6}, {1, 2, 5, 4, 3, 6}, {3, 2, 5, 4, 1, 6}, {3, 2, 1, 4, 5, 6}, {5, 2, 1, 4, 3, 6}, and {5, 2, 3, 4, 1, 6}

Six permutations.

(b) Exactly three integers are in their natural position.

Exactly 3 integers leaves us with permutations between three numbers, so 3! = 6.

We have six permutations.

(c) 1 and 6 are in their natural positions

If only 1 and 6 are in there natural position, then we have permutations between 4 elements.

We have 4! Permutations.

4! = 4×3×2×1 = 24

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step-by-step explanation:

answer: a video game has three difficulty levels: easy, medium, and expert. the game's scoring works in such a way that a player has to score a fixed number of points to advance to the next level, and then the scoring starts over. the total score needed to clear all three difficulty levels is 7,500. two times the number of points needed to clear the easy level is 500 more than the number of points needed to clear the medium level. twice the number of points needed to clear the medium difficulty level is 1,500 more than the number of points needed to clear the expert level. if x, y, and z represent the easy, medium, and expert levels, respectively, and a is the coefficient matrix of the system of equations modeling this situation, identify the inverse matrix, a-1, and the solution matrix, x, of the system of equations.

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step-by-step explanation: