SOLUTION: how many permutations can be made from the letters in the word "MONDAY" if a). all letters are used b). arranged where A is the first letter c). arranged if A, N, and D, are t

Algebra ->  Permutations -> SOLUTION: how many permutations can be made from the letters in the word "MONDAY" if a). all letters are used b). arranged where A is the first letter c). arranged if A, N, and D, are t      Log On


   

Question 1196163: how many permutations can be made from the letters in the word "MONDAY" if
a). all letters are used
b). arranged where A is the first letter
c). arranged if A, N, and D, are together
d). arranged if A, N, and D, are seperated
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Part (a)

There are 6 letters in the word "MONDAY"
If we scramble them up, then we have...
6 ways to pick the first letter
5 ways to pick the second letter
4 ways to pick the third letter
and so on until counting down to 1 way to select the final letter

6*5*4*3*2*1 = 720 different ways to scramble those six letters.
We can shorten this to 6! = 720
The exclamation mark indicates a factorial.

Answer: 720

=============================================================

Part (b)

Letter "A" is locked in the first slot
We have 6-1 = 5 remaining letters to scramble up.

There are 5 choices for the second slot
Then 4 choices for the third
etc etc

5*4*3*2*1 = 120

Answer: 120

=============================================================

Part (c)

Let X stand in place of "AND", where those three letters are presented in any order.
The word "MONDAY" becomes "MOXY"

We have 4! = 4*3*2*1 = 24 ways to arrange the letters in "MOXY"

Then for any of those 24 permutations, there are 3! = 3*2*1 = 6 ways to arrange the letters "A", "N", "D".

Ultimately there are 6*24 = 144 ways to arrange the letters in "MONDAY" such that "A", "N", "D" stick together in any order
Examples:
MONDAY
MODNAY
MONADY

Answer: 144

=============================================================

Part (d)

I interpret this to mean that we want "A", "N", "D" to be three isolated islands.
Meaning that no two letters from the set {"A", "N", "D"} can be adjacent.

Something like "MONDYA" wouldn't work since "N" and "D" are neighbors.
But something like "MNODYA" would work

It might help to think of the other letters as blank spaces
_N_D_A
In this configuration, we have 3! = 6 ways to arrange the letters {M,O,Y} and 3! = 6 ways to arrange the letters {A,N,D}. That gives 3!*3! = 6*6 = 36 ways to have this current template.


We could also have something like this
N_D_A_
There's nothing special about this since we also get 3!*3! = 36 ways to arrange the letters here.

Another template is this:
N_,_D_A
the comma is to show there are two blank spaces instead of 1
We will also have 3!*3! = 36 ways to arrange the letters in this pattern.

Yet another template:
N_D_,_A
There's also 36 ways to arrange the letters in this format.

There were 4 templates mentioned. Each having 36 ways to arrange the letters. That means we have 4*36 = 144 ways to have the letters in any of the templates mentioned where the letters "A", "N", "D" are separated. No two such letters can be next to one another.

Answer: 144

=============================================================

I confirmed each answer with this calculator here:
https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html
This calculator shows the numeric count of a permutation or combination. It also lists out the actual permutations or combinations themselves.

For example, if you had the set A,B,C then it will show the 6 permutations of...
ABC
ACB
BAC
BCA
CAB
CBA
The order of the permutations may be different from what I've listed.