# In how many ways can the letters of the english alphabet be arranged s

In how many ways can the letters of the english alphabet be arranged so that there are seven letter between the letters A and B, and no letter is repeated?

I have searched this question and have seen many interpretations.

Ans: 24 x 23 x 22 x21 x20 x19 x 18

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Ans 24P7 * 2 * 18

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My solution

since there are 36 positions for A and B (1-9, 9-1, 2-10,10-2, ...18-26,26-18) and remaining 24 letters can be positioned in 24! ways, to me, the third answer 24!*36 appears right. Please tell if this is correct.

• In your solution you meant 36 positions for A and B. – user21820 Mar 2 '16 at 9:54
• @user21820 thx, it was a typo, corrected – Kiran Mar 2 '16 at 9:55

• Choose a place for A and put B appropriately ahead: $26-(7+1)$ options
• Reorder A and B in every possible way: $2!$ options
• Choose $7$ letters out of the letters between C and Z: $\binom{26-2}{7}$
• Reorder those $7$ letters in every possible way: $7!$ options
• Reorder the remaining letters in every possible way: $(26-2-7)!$ options
The answer is therefore: $(26-(7+1))\cdot2!\cdot\binom{26-2}{7}\cdot7!\cdot(26-2-7)!=24!\cdot36$