Permutations- How many three letter words can you produce from the letters BALLSY?

Question

I assumed the way to tackle this problem would be $\frac{n!}{(n-r!)(2!)}$ to account for the two l's

which would result in

$\frac{6!}{3!2!}=60$

However, when I enumerate the different possibilities, I end up with over 60 possibilities- see below:

BAL LSY LBY LAY LAS BLS ASY BAY ABS BYS ALL

ABL LYS YBL LYA LSA SLB SAY BYA SAB BSY LLA

ALB YLS BLY YLA SLA LBS SYA YBA BAS YSB LAL

BLA SLY YLB ALY ALS LSB YSA ABY BSA SYB BLL

LBA SYL BYL AYL SAL BSL YAS YAB ASB SBY LBL

LAB YSL LYB YAL ASL SBL AYS AYB SBA YBS LLB... and so on

How can I calculate this using the permutation formula?

Answer 1

Nice question. I would first ignore the double "L" and just compute the ways to permute $3$ out of the $6$ numbers. Using permutations, this is simply $\frac{6!}{3!} = 6 * 5 * 4 = 120.$

Now I account for overcount of the double "L." Notice that I do this after counting the total because those permutations without any L's (like "BAS") are NOT overcounted. We notice that there are $\dbinom{4}{1} * 3 = 12$ permutation forms that contain $2$ L's and $1$ other letter - we first pick a non-L letter, then arrange this letter and two L's into a three-letter permuation (we just have to pick a spot for the non-L letter). We counted all $12$ of these twice. So we need to subtract $12$ from our original count. Now what about $1$ L and two other letters? For this, we notice that there are $\dbinom{4}{2} * 3 * 2 = 36$ permutation forms - we first select two non-L letters from the four available, then arrange these two and the L into a three-letter permutation, which is why we multiply by $3 * 2.$ We have overcounted $12 + 36 = 48$ times.

Our answer is $120 - 48 = \boxed{72}.$ The takeaway: carefully address your overcounts. Using combinations may also be helpful here. Hope this helps!

Answer 2

For words that contain at most one L, there are $5!/2! = 60$ ways to permute.

For words that contain two Ls, there are 12 ways to permute

Answer 3

Number of words with at most 1 L = number of words that can be made from BALSY = 5*4*3 = 60.

Number of words with made with 2 Ls = number of ways to arrange LL and one of BASY = 4*3 = 12. (4 choices for the other letter, and 3 places to put it.)

Number of words with either 0, 1 or 2 Ls = 60 + 12 = 72.