1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
salantis [7]
2 years ago
15

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

Mathematics
1 answer:
NARA [144]2 years ago
6 0

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

\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\}

Then

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}\}\\

Note that since

F \cap R=\varnothing, Perm(F)\cap Perm(R)\ne \varnothing

But since

B \cap R \ne \varnothing, Perm(B)\cap Perm(R)= \varnothing

and

B \cap F \ne \varnothing , Perm(B)\cap Perm(F)= \varnothing

Since

|\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!\\

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

and

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

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

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

This link contains another solved problem on permutations:

brainly.com/question/7951365

You might be interested in
What is the measure of angle C?
Aleksandr-060686 [28]

Answer:

38 or 76 but i think its 76

Step-by-step explanation:

3 0
3 years ago
A.8.4 t=_______kg
Art [367]
<h2><u>ANS</u></h2>

A) 8400 kg

B) 10 km

C) 0.231 kg

D) 6000000 g

E) 0.007 t

F) 0.00125 kg

G) 3900 kg

H) 2 kg

I) 5000 g

J) 3.6 kg

<h3><u>T</u><u>hank</u> <u>You</u> !!!</h3>

7 0
3 years ago
Rewrite the fractions 3/4 and 5/16 as fractions with a least common denominator.
nata0808 [166]

Answer:

12/16 and 5/16

Step-by-step explanation:

We are given two fractions with denominator 4 and 16.

To find the least common denominator, of 4 and 16.

Factorise 4 and 16.

4 = 2x2  

16 = 2x2x2x2

i.e. 4 divides 16.

Hence  Least common denominator = bigger number = 16

Make the given fractions to equivalent fractions with common denominator 16.

3/4 = 4(3)/4(4) = 12/16 and

5/16 = 1(5)/16(1)= 5/16

Hence answer would be

12/16 and 5/16


6 0
3 years ago
Read 2 more answers
100 cups of gasoline during a drive. he paid $3.12 per gallon for gas. How much did the gas cost?
Vanyuwa [196]
There are 16 cups in a gallon. Divide 100 into 16 to get 6.25. You now just multiply 3.12 by 6.25 to get 19.5. Your answer is $19.50.

hope this helps!
3 0
3 years ago
Read 2 more answers
Is 0.032 a terminating decimal
svp [43]
I don’t think it is, sorry I’m not 100% sure, good luck though
3 0
3 years ago
Other questions:
  • A coin is tossed 2 times. find the probability that all 2 tosses are tails
    15·1 answer
  • Susie buys 1/2 lb of tomatoes for $1.99, what is the unit rate of the tomatoes?
    6·2 answers
  • What is 2,548 rounded to the<br> nearest hundred?
    7·2 answers
  • 2. How far will a bird have traveled in 13 seconds if it maintained a speed of 31 m/s?
    9·2 answers
  • A moving company needs to replace a circualr glass mirror that it broke during a move. The dimensions of a piece of the broken m
    12·1 answer
  • A father lifts a toddler 1.5 m up in the air. The child gains 187.5 J of energy in her gravitational potential energy store as a
    14·1 answer
  • Hellllllllllllllllllllllllpppppppppppppppppppppppppp
    13·1 answer
  • HELP ME PLEASE NOBODY IS ACTUALLY HELPING THIS OS SO HARD AND IM STUCK 7TH GRADE MATH
    8·2 answers
  • Jordan buys a notebook for himself and
    12·1 answer
  • Find the value of x that makes m ∥ n
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!