If n is a positive integer, how many 5-tuples of integers from 1 through n can be formed in which the elements of the 5-tuple are written in increasing order but are not necessarily distinct? In other words, how many 5-tuples of integers (h, i, j, k, m) are there with 1 ≤ h ≤ i ≤ j ≤ k ≤ m ≤ n? As in Example 9.6.3, you can represent any ordered 5-tuple of integers (h, i, j, k, m) with 1 ≤ h ≤ i ≤ j ≤ k ≤ m ≤ n as a string of n − 1 vertical bars and 5 crosses, with the position of crosses indicating which 5 integers from 1 to n are included in the 5-tuple. Thus, the number of 5-tuples is the same as the number of strings of n+4 vertical bars and 5 crosses, which is n(n+1)(n+2)(n+3)(n+4) 120​ .

Step-by-step explanation:

Given

5 tuples implies that:

implies that:

Required

How many 5-tuples of integers are there such that

From the question, the order of the integers h, i, j, k and m does not matter. This implies that, we make use of combination to solve this problem.

Also considering that repetition is allowed:  This implies that, a number can be repeated in more than 1 location

So, there are n + 4 items to make selection from

The selection becomes:

Expand the numerator

Solved

the graph in the attached figure

step-by-step explanation:

we have

using a graphing tool

the graph in the attached figure

the roots are ( x-intercepts) x=-0.5,x=1,x=3

the y-intercept is the point (0,3)

if you are looking for the unit rate then the answer would be 9 because unit rate is distance divided by time.

step-by-step explanation: