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
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
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