Find the number of ways to arrange all the letters in the word “MALAYSIA” if the first letter must be a consonant and the last letter must be a vowel.

2,593

Solution 1

There are $4$ ways for a consonant to be in the first position, and $4$ ways for the last letter to be a vowel. Two letters are now taken, so there are $6!$ ways to arrange the remaining 6 letters. This gives $4\cdot4\cdot6!=11520$ permutations as your answer if the three As are not treated as identical. If they are, then there are $3!$ ways to arrange them, so divide to get $\dfrac{11520}{3!}=1920$ as your answer.

Solution 2

we have 8 letters in total (_ _ _ _ _ _ _ _ _)

4 vowels - A,A,A,I

4 consonants - M,L,Y,S

First letter must be consonant so we can pick any letter from M,L,Y,S so we have 4 options - (4 _ _ _ _ _ _ _)

Last letter must be a vowel we can pick A or I so only 2 choices

  • choosing A
  • choosing I

Choosing A

we are left with 6 letters 3 consonants and 3 vowels (A,A,I)

so number of possibilities are (6p6)/2!

Choosing I

we are left with 6 letters 3 consonants and 3 vowels (A,A,A)

so number of possibilities are ($\frac{^6p_6}{3!}$)

so total is 4*(6p6/2! + 6p6/3!) = 1920

Your first question answer should be 6720

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Updated on August 19, 2022

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  • Nasser Ali adam
    Nasser Ali adam about 1 year

    Find the number of ways to arrange all the letters in the word “MALAYSIA” if the first letter must be a consonant and the last letter must be a vowel.

    well here is the way i tried to answer it : we have 4 consonants and 4 vowels so i came up with this formula 16($^6 P_6$)=11520 now i am not sure if this is the correct answer or the correct way to answer the question .