Math, asked by Shaheensheikh4650, 10 months ago

Determine the number of arrangements of letters of the word algorithm if a) vowels are always together b) no two vowels are together c) consonants are at even positions d) o is the first and T is the last letter

Answers

Answered by prettystefina11
53

Answer:

(a) 30240, (b) 332640, (c) 0, (d) 5040

Step-by-step explanation:

Given word;

ALGORITHM

Number of letters = 9

Number of vowels = 3 (A, O, I)

Number of consonants = 6 (L, G, R, T, H, M)

(a) Vowels are always together:

There are 9 positions like;

1 2 3 4 5 6 7 8 9

The vowels should be together.

Their positions could be;

1 2 3

2 3 4

.

.

.

7 8 9 = 7 ways

The vowels can be arranged in themselves in 3! Ways.

The remaining consonants can be arranged in 6! Ways in 6 remaining positions.

Therefore,

The number of arrangement of letters if the vowels need to be together = 7 x 3! x 6! = 7 x 3 x 2 x 6 x 5 x 4 x 3 x 2 x 1 = 30240

(b) No two vowels are together:

We can solve this by finding out the total kinds of arrangements as a whole and subtracting the result in (a)

Total number of ways the letters can be arranged = 9! = 362880

The number arrangements if the vowels need to be together = 30240

The number of arrangements if no two vowels are together = Total number of arrangements — Arrangements when vowels are together

= 362880 — 30240

= 332640

(c) Consonants are at even position:

There are 6 consonants.

The number of positions = Number of letters = 9

Like;

1 2 3 4 5 6 7 8 9

There are 4 even portions 2 4 6 8

There are 6 consonants. If 4 of the consonants could be placed in even positions, the remaining 2 should be placed in odd positions.

Placing all the consonants at even positions is not possible.

So, the number of arrangements with consonants at event positions = 0

(d) O is the first and T is the last letter:

First letter is O

Last letter is T

The remaining 7 positions can be filled with the remaining 7 letters in 7! Ways

Therefore, the number of arrangements with O as the first letter and T as the last letter = 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1

                                              = 5040

Answered by Anonymous
5

Answer:

Step-by-step explanation:

Answer:

(a) 30240, (b) 332640, (c) 0, (d) 5040

Step-by-step explanation:

Given word;

ALGORITHM

Number of letters = 9

Number of vowels = 3 (A, O, I)

Number of consonants = 6 (L, G, R, T, H, M)

(a) Vowels are always together:

There are 9 positions like;

1 2 3 4 5 6 7 8 9

The vowels should be together.

Their positions could be;

1 2 3

2 3 4

.

.

.

7 8 9 = 7 ways

The vowels can be arranged in themselves in 3! Ways.

The remaining consonants can be arranged in 6! Ways in 6 remaining positions.

Therefore,

The number of arrangement of letters if the vowels need to be together = 7 x 3! x 6! = 7 x 3 x 2 x 6 x 5 x 4 x 3 x 2 x 1 = 30240

(b) No two vowels are together:

We can solve this by finding out the total kinds of arrangements as a whole and subtracting the result in (a)

Total number of ways the letters can be arranged = 9! = 362880

The number arrangements if the vowels need to be together = 30240

The number of arrangements if no two vowels are together = Total number of arrangements — Arrangements when vowels are together

= 362880 — 30240

= 332640

(c) Consonants are at even position:

There are 6 consonants.

The number of positions = Number of letters = 9

Like;

1 2 3 4 5 6 7 8 9

There are 4 even portions 2 4 6 8

There are 6 consonants. If 4 of the consonants could be placed in even positions, the remaining 2 should be placed in odd positions.

Placing all the consonants at even positions is not possible.

So, the number of arrangements with consonants at event positions = 0

(d) O is the first and T is the last letter:

First letter is O

Last letter is T

The remaining 7 positions can be filled with the remaining 7 letters in 7! Ways

Therefore, the number of arrangements with O as the first letter and T as the last letter = 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1

= 5040

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