Prove using mathematical induction that the number of permutations of the set {1, 2, . . . , n} with n elements is n!, for natural number n ≥ 1. As an example, the permutations of {1, 2, 3} are {[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]}.

Step-by-step explanation:

We willl proof the statement using induction.

REcalll that to prove something using induction, we must prove it for a base case, then, we assume the result for n, and prove it for the case n+1.

Consider our base case as n=2, since n=1 is trival.

Note that given the set {1,2} the posible permutations are [1,2] and [2,1] (2 permutations). Recall that 2! = 1*2 = 2. So, the result applies to the base case.

Suppose that the result holds for n. That is, given a set {1,...,n} of n elements, the number of permutations is n!

Consider the case in which you have a set of n+1 elements {1,...,n,n+1}.

The total number of permutations of this set can be constructed as follows. Consider one permutation of the set {1, ...., n}. For illustration, consider the identity permutation. That is,

1, 2, 3, 4, ...., n. Recall that a permutation of the whole set can be constructed by taking one permutation of the set {1,...,n} and placing the element n+1. So, in our case, this are two examples of permutations of the new set :

n+1, 1,2,3,4,....,n.

1,n+1,2,3,4,...,n.

So, our problem reduces to count the total posibilities where we can place the last element n+1. Note, that we can place the last element in exactly n+1 positions to get n+1 permutations.

That means, that given one permutation of the set {1,...,n} we get n+1 permutations of the set {1,...,n+1}. Hence the total number of permutations of the set {1,...,n,n+1} is n+1 times the number of permutations of the set {1,...,n}. Using the induction hypothesis, we have that the number of permutations of the set {1,...,n,n+1} is

So, the result holds for any n.  

- "runner b is moving at a slower rate."

- "the graphs shown are graphs of distance(km) against time(min).

the slopes of this graphs are are given by:

slope = (change in y)/(change in x)

= (change in distance)/(change in time)

this is the formula for finding the speed. so, the slope of these graphs represent the speed of the runner.

taking each division in the graph to represent 1 on both axis:

runner a ⇒ speed = (2-0)/(1-0)

= 2/1

= 2 km/min

runner b ⇒ speed = (3-0)/(3-0)

= 3/3

= 1 km/min

this shows that   runner b is moving at a slower rate."

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step-by-step explanation:

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