# How many different words can be formed with the letters of 'SKYSCRAPERS' if A and E are together?

a. How many different words can be formed with the letters of 'SKYSCRAPERS' if A and E are together?

b. How many of the words formed from 'SKYSCRAPERS' begin with 'K' and end with 'A'?

Attempt:

I tried solving the a part using $9! \cdot 10C2$ since we can arrange AE among the other 9 words in 10C2 ways but I don't think my attempt is right.

If I am right, I plan to solve the b part using the same method.

This combinatorics question is from my textbook. I came across it when solving different questions. Please assist, if I am wrong with the two parts. Thanks

• For part $b$, notice that this is exactly the same as "how many different words can be formed with the letters of 'SYSCRPERS'", because the 'K' and 'A' are fixed in place. – user137731 Oct 10 '16 at 3:35
• @Bye_World thank you very much for that. The b part will now be 30,240. I didn't look at it that way. – john scott Oct 10 '16 at 3:40
• @Bye_World how about the part a. Is that right? – john scott Oct 10 '16 at 3:40

The word SKYSCRAPERS has eleven letters. If we place the letters A and E in a box, we have ten objects to arrange. They include $3$ S's, $1$ K, $1$ Y, $1$ C, $2$ R's, $1$ P, and $1$ box containing A and E. Choose three positions of the ten positions for the S's, which can be done in $\binom{10}{3}$ ways, and two of the remaining seven positions for the R's, which can be done in $\binom{7}{3}$ ways. We are left with five positions in which to arrange five different objects, the four single letters and the box containing A and E, which can be done in $5!$ ways. Finally, arrange A and E within the box, which can be done in $2!$ ways. Hence, the number of arrangements of the letters of SKYSCRAPERS in which A and E are adjacent is $$\binom{10}{3}\binom{7}{2}5!2! = \frac{10!}{7!3!} \cdot \frac{7!}{5!2!} \cdot 5! \cdot 2! = \frac{10!}{3!}$$
As @Bye_World pointed out in the comments, since the positions of K and A are fixed, this amounts to arranging the nine letters of the word SYSCRPERS. Of those nine letters, there are $3$ S's, $1$ Y, $1$ C, $2$ R's, $1$ P, and $1$ E. Proceeding as above, we choose three of the nine positions for the S's, two of the remaining six positions for the R's, then arrange the four single letters in the remaining four positions, which can be done in $$\binom{9}{3}\binom{6}{2}4! = \frac{9!}{6!3!} \cdot \frac{6!}{4!2!} \cdot 4! = \frac{9!}{3!2!}$$ ways. The factor of $3!$ in the denominator represents the number of ways the three S's can be permuted among themselves within a given arrangement, and the factor of $2!$ in the denominator represents the number of ways the two R's can be permuted among themselves within a given arrangement. Since permuting indistinguishable letters among themselves within a given arrangement does not produce an arrangement that is distinguishable from the given arrangement, we would be counting each distinguishable arrangement $3!2! = 6 \cdot 2 = 12$ times if we did not divide by this factor.