Question #178356

A word is to be formed using some or all of the 10 letters.

M A N C H E S T E R

Find the total number of ways of forming the word if

a) The word is 10-letter long. The letters can be in any combinations. (2 marks)

b) The word is 10-letter long. The word must begin with a consonant and end with a vowel. (5 marks)

c) The word is 2-letter long, and must not include T or E. (2 marks)

d) The word is 7-letter long. The word must contain at least one E. (6 marks)

e) The word is 5-letter long, no other restrictions. (5 marks)

Expert's answer

a) The wanted number of ways is equal to the number of permutations of 10 elements with the repetition of two elements (since this word contains two identical letters E).

Then .

Answer: 1814400 ways.

b) 1 consonant letter from a given word can be selected in 7 ways. 1 vowel from a given word can be selected in 3 ways (two vowels are repeated). The remaining 8 letters can be rearranged by 8! ways.

Then

Answer: 564480 ways.

c) The first letter can be selected in 7 ways, the second in 6 ways. Then

Answer: 42 ways

d) Find the number of words with one letter E

Since the letter E can be in any of the 7 places, and the remaining 6 letters can be selected in ways then number of words with one letter E is equal to

Find the number of words with two letters E.

Places for the letters E can be chosen in ways, the remaining 5 letters can be selected in ways. Then then number of words with 2 letters E is equal to .

Then total number is 141120+141120=282240.

Answer: 282240 ways.

e) Since two letters E are repeated, the wanted number is

Answer: 15120 ways.

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