
Dear Sir,
please help me with the following doubt.
Q. In how many ways the letters in the word MATHEMATICS can be arranged if vowels are to be arranged at extreme ends?
Ans. Now here, the vowels are A, A, E, I that are to be arranged at ends. So, these can be arranged in ^{4}P_{2}/2! (using the concept 4 things are to be arranged at 2 places and out of these 2 are same)
Also, rest 9 letters can be arranged in 9!/2! 2! ways.
So, total number of ways will be ( ^{4}P_{2}/2! ) (9!/2! 2!) My doubt is ^{4}P_{2}/2! will give 6. However, when I write the combination then it comes out to be 7. Please tell me what is the mistake that i have done if I use formula rather than writing the combinations manually.
A...............A, A...............E, A...............I, E...............A, E...............I, I................E, I................A
Regards, Rupali
