Mathematics, 16.09.2019 08:30 Hrjohnson2004

How many sets of three integers between 1 and 20 are possible if no two consecutive integers are to be a set?

Answers: 1

Show answers

Answers

Answer from: Theyoungin420

The question is an illustration of combination (i.e. selections)

There are 816 sets of three non-consecutive integers between 1 and 20

The given parameters are:

$\mathbf{N = 20}$ --- number of integers

$\mathbf{r = 3}$ --- the set of integers

For the integers to be selected such that there are no two consecutive integers.

The value of n must be:

$\mathbf{n = N - r + 1}$

So, we have:

$\mathbf{n = 20 - 3 + 1}$

$\mathbf{n = 18}$

The number of selection is then calculated as:

$\mathbf{^nC_r = \frac{n!}{(n - r)!r!}}$

So, we have:

$\mathbf{^{18}C_3 = \frac{18!}{(18 - 3)!3!}}$

$\mathbf{^{18}C_3 = \frac{18!}{15!3!}}$

Expand

$\mathbf{^{18}C_3 = \frac{18 \times 17 \times 16 \times 15!}{15! \times 3 \times 2 \times 1}}$

$\mathbf{^{18}C_3 = \frac{18 \times 17 \times 16}{3 \times 2 \times 1}}$

$\mathbf{^{18}C_3 =3 \times 17 \times 16}$

$\mathbf{^{18}C_3 =816}$

Hence, there are 816 sets of three non-consecutive integers between 1 and 20

Read more about combinations (selection) at:

link

Answer from: animegirl02

There are $\dbinom{20}3=1140$ total possible ways to pick any three integers from the set.

Of the total, there are 18 consisting of consecutive triplets $(\{1,2,3\},\{2,3,4\},\ldots,\{18,19,20\})$ .

Now, of the total, suppose you fix two integers to be consecutive. There would be 19 possible pairs $(\{1,2\},\{2,3\},\ldots,\{19,20\})$ , and for each pair 18 possible choices for the third integer (for instance, $\{1,2\}$ can be taken with 3, 4, ..., 20), to a total of $19\times18=342$ . To avoid double-counting (e.g. $\{1,2\}$ can't go with 3; $\{2,3\}$ can't go with 1 or 4), we subtract 1 from the extreme pairs $\{1,2\}$ and $\{19,20\}$ (twice), and 2 from the rest (17 times).

So, the number of triplets that don't consist of pairwise consecutive integers is

$1140-(18+342-(2\times1+17\times2))=816$

I don't know how useful this would be to you, but I've verified the count in Mathematica:

In[8]:= DeleteCases[Subsets[Range[1, 20], {3}], x_ /; x[[2]] == x[[1]] + 1 || x[[3]] == x[[2]] + 1] // Length
Out[8]= 816

Answer from: Quest

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.

ask for details followreport by buttfacelily 2 minutes ago

answers

rauschg

rauschg · hand

know the answer? add it here!

step-by-step explanation:

Answer from: Quest

3+x

step-by-step explanation:

Another question on Mathematics

Mathematics, 21.06.2019 13:00

What is the least common multiple of 16 and 3?

Answers: 2

Answer

Mathematics, 21.06.2019 19:00

Rena is building a 1: 180 scale model of a real castle. her model has a rectangular base that is 3 feet wide and 4 feet long what is the area of the base of the actual castle in square feet

Answers: 2

Answer

Mathematics, 21.06.2019 20:30

Select all the expressions that will include a remainder.

Answers: 1

Answer

Mathematics, 21.06.2019 21:30

Jude is making cement for a driveway. the instructions show the amount of each ingredient to make 1 batch of cement. complete each statement to adjust the ingredients for each new situation if jude uses these instructions.

Answers: 3

Answer

You know the right answer?

How many sets of three integers between 1 and 20 are possible if no two consecutive integers are to...

Questions