How many permutations of the 26 letters of the English alphabet do not contain any of the strings fish, rat, or bird

The number of permutations of the 26 letters of the English alphabet that do not contain any of the strings fish, rat, or bird is 402619359782336797900800000

Let

[tex]\mathcal{E}=\{\text{All lowercase letters of the English Alphabet}\}\\\\B=\overline{\{b,i,r,d\}} \cup \{bird\}\\\\F=\overline{\{f,i,s,h\}} \cup \{fish\}\\\\R=\overline{\{r,a,t\}} \cup \{rat\}\\\\FR=\overline{\{f,i,s,h,r,a,t\}} \cup \{fish,rat\}[/tex]

Then

[tex]Perm(\mathcal{E})=\{\text{All orderings of all the elements of } \mathcal{E}\}\\\\Perm(B)=\{\text{All orderings of all the elements of } \mathcal{E} \text{ containing bird}\}\\\\Perm(F)=\{\text{All orderings of all the elements of } \mathcal{E} \text{ containing fish}\}\\\\Perm(R)=\{\text{All orderings of all the elements of } \mathcal{E} \text{ containing rat}\}\\\\Perm(FR)=\{\text{All orderings of all the elements of } \mathcal{E} \text{ containing both fish and rat}\}\\[/tex]

Note that since

[tex]F \cap R=\varnothing[/tex], [tex]Perm(F)\cap Perm(R)\ne \varnothing[/tex]

But since

[tex]B \cap R \ne \varnothing[/tex], [tex]Perm(B)\cap Perm(R)= \varnothing[/tex]

and

[tex]B \cap F \ne \varnothing[/tex] , [tex]Perm(B)\cap Perm(F)= \varnothing[/tex]

Since

[tex]|\mathcal{E} |=26 \text{, then, } |Perm(\mathcal{E})|=26! \\\\|B|=26-4+1=23 \text{, then, } |Perm(B)|=23!\\\\|F|=26-4+1=23 \text{, then, } |Perm(F)|=23!\\\\|R|=26-3+1=24 \text{, then, } |Perm(R)|=24!\\\\|FR|=26-7+2=21 \text{, then, } |Perm(FR)|=21!\\[/tex]

where [tex]|Perm(X)|=\text{number of possible permutations of the elements of X taking all at once}[/tex]

and

[tex]|Perm(F) \cup Perm(R)| = |Perm(F)| + |Perm(R)| - |Perm(FR)|\\= 23!+24!- 21! \text{ possibilities}[/tex]

What we are looking for is the number of permutations of the 26 letters of the alphabet that do  not contain the strings fish, rat or bird, or

[tex]|Perm(\mathcal{E})|-|Perm(B)|-|Perm(F)\cup Perm(R)|\\= 26!-23!-(23!+24!- 21!)\\= 402619359782336797900800000 \text{ possibilities}[/tex]

This link contains another solved problem on permutations:

https://brainly.com/question/7951365

Dan was thinking of a number. Dan doubles it and adds 0.8 to get an answer of 58.9. Form an equation with
x
from the information.

8. Makayla and Haylee are selling oranges for a school fundraiser. Customers can buy small boxes of oranges and large boxes of oranges. Haylee sold 6 small boxes and 12 large boxes for a total of $330. Makayla sold 12 small boxes and 4 large boxes for a total of $280. a. Write a system of equations that gives the cost of one small box of oranges and one large box of oranges.

b. Find the cost of a small box of oranges and the cost of a large box of oranges​

if f(x) = (3+x)/(x-3), what is f(a+2)​

7.4{5[2 +16 ÷ (4× 2) − 3]− 4}− 24 =

solve for x

3x/6 + 1 = 7

i’m physically begging for someone to help i do not understand

What is the answer please?

what is the answer???????​

answer..............................

What can you multiply by to find:

50% of a number?
10% of a number?
75% of a number?

What is the 5th equivalent fraction to 1/12?

Ali walks 256km and Ahmed walks 112km. How much did they walk altogether? Give answer as a mixed number in its lowest term

The ratio of roses to tulips is 2:3. What percent of the flowers are roses?

A company spent $2415 on a plane ticket for seven employees all the tickets cost the same amount how much did each ticket cost

Solve for x
5x – 1 < 2
Give your answer as a fraction in its simplest form.

Pls help I’ve got no idea

There are 35 students in the elementary band. The band is doing a fundraiser, and each student must sell 12 items. How many items will be sold altogether? Use a different multiplication strategies in each of the short answer questions.

d = √ ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 Evaluate d if ( x 1 , y 1 ) = ( − 3 , 4 ) and ( x 2 , y 2 ) = ( − 2 , 4 ) .

Factorise:
a) 22y -33z b) ax + bx –ay – by

How to calculate percentile.

William bought 6 shirts for $119.70. How much did one shirt cost him?

Two trains leave a train station traveling on parallel and adjacent tracks. Train A is traveling due north at a constant speed of 76 miles per hour. Train B is traveling due south at a constant speed of 44 miles per hour. If both trains leave the station at the same time, in approximately how many minutes will the trains be 22 miles apart? O 11 O 18 O 60 o 98​

f (x)=x^2 what is g (x)

4. Solve for x, and explain which theorem or
postulate that helps you find the answer.

PLEASE HELP ME WITH THESE MATH PROBLEMS!! DUE TONIGHT!!

I got the answer but I don't know how to show my work, can ya'll tell me what I should write??

How to find where inflection points are.

Simplify: square root of 18x^7

O 9x^3 square root x
O 3x^3 square root 2x
O 3x square root x^3

You are given right triangle that has a perimeter of 42.56m, a hypotenuse of 21.34m and a base length of 8.76m. Find the area of the right triangle.

One cup of coffee needs 2 spoons of milk. If one bottle of milk has 50 spoons, write an inequality representing the possible cups of coffee one bottle of milk can make.

order -9, -16, -13, 0 from least to greatest