How many ways are there to arrange the letters in the word Tennessee?

9 total words, 4 "E" , 2 "N" , 2 "S"

= 9! / 4! 2! 2! = 3780. Answer C

I used the standard textbook method:

The basic formula to find out the number of total arrangements is \(x!\), therefore Tenessee could be arranged in \(9!\) ways. But the word 'Tennessee' consists of 2 letters of 'n's, 2 letters of 's's and 4 letters of e's. So, we need to divide the total number of arrangements by \(2!\) (to account for the 'n's) \(2!\) (to account for the 's's) and \(4!\) (to account for the 'e's). Therefore \(\frac{9!}{2!2!4!}\) \(=\) \(\frac{9*8*7*6*5}{4}\) \(= 9*7*6*5*2 = 3780\). Out of curiosity, is there any other way we could solve this problem?...please let me know.

Please consider giving me KUDOS if you felt this post was helpful! Thanks.

VERITAS PREP OFFICIAL SOLUTION:

Correct Answer: (C)

Remember your formula for Permutations with Repeating Elements: N!/A!B!..., such that N is the total number of letters to be arranged and A, B, and any other values in the denominator are the number of each repeating element that you have. In our case the formula could be expressed as 9!/2!4!2!, where 9 is the total number of letters, 2 the number of N’s, 4 the number of E’s, and the other 2 the number of S’s. From here, simplify. The fraction can be rewritten as (9 * 8 * 7 * 6 * 5) / 2 * 2, or (9 * 7 *6 * 5 * 2). From here, multiply quickly: 9 * 42 * 10 = 9 * 420 = about 3600, or answer (C).

If the letters in Tennessee were all different, then we could arrange the letters in 9! ways. However, there are 4 occurrences of the letter e and 2 occurrences each of the letters n and s. We use the indistinguishable permutations formula, which requires that we divide by the factorial of each of the number of indistinguishable letters. Thus, the number of ways one can arrange the letters in Tennessee is:

9!/(4!2!2!) = 3780

Answer: C

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.