Question

The number of ways in which all the letters of the word 'TANATAN' can be arranged so that

no two alike letters are together

(A) 24 (B) 36 (C) 38 (D) 40

Answer 1

I hope answer is 38
I am not sure

Answer 2

According to permutation, we can find out that the number of ways in which all the letters of the word 'TANATAN' can be arranged so that no two alike letters are together is 60.

It is given to us that -

The word is 'TANATAN'

We have to find out the number of ways in which all the letters of the word 'TANATAN' can be arranged so that no two alike letters are together.

Now, in the word  'TANATAN',

total number of letters = 7

total number of 'T' letters = 2

total number of 'A' letters = 3

total number of 'N' letters = 2

So, the total number of ways or arranging 'TANATAN' =

$\frac{7!}{2!3!2!} \\= \frac{7*6*5*4*3*2*1}{2*1*3*2*1*2*1}\\= 210$ ------ (1)

Now, if we take two alike 'T' letters, the total number of ways or arranging 'TANATAN' =

$\frac{6!}{3!2!}\\ = \frac{6*5*4*3*2*1}{3*2*1*2*1} \\=60$------ (2)

Now, if we take three alike 'A' letters, the total number of ways or arranging 'TANATAN' =

$\frac{5!}{2!2!} \\= \frac{5*4*3*2*1}{2*1*2*1} \\=30$ ------- (3)

Now, if we take two alike 'N' letters, the total number of ways or arranging 'TANATAN' =

$\frac{6!}{2!3!} \\= \frac{6*5*4*3*2*1}{2*1*3*2*1} \\=60$ ------- (4)

Thus, the number of ways in which all the letters of the word 'TANATAN' can be arranged so that no two alike letters are together =

210 - (60 + 30 + 60) [From equations (2), (3), (4)]

= 210 - 150

= 60

Therefore, through permutation, we can find out that the number of ways in which all the letters of the word 'TANATAN' can be arranged so that no two alike letters are together is 60.

To learn more about permutations visit https://brainly.com/question/27058178

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