Hello! I wrote my final exam yesterday and this question had me stumped:

How many ways can you arrange the letters in the word GOOGLE where all the consonants stay together?

I tried many different ways of figuring it out and ended up with many different answers! My brain tells me that, without restrictions, the number of arrangements is 6! = 720. Therefore my answer cannot exceed this number. Next I think that the consonants form one group that can be rearranged 3! Then, the vowels form another 3 separate groups. At the end, you would divide out the repeats (2 g's and 2 o's), correct?

I ended up choosing the answer of 90. I can't recall how I got this number and I have no idea if it's right. Can someone provide the answer to this question and the steps they took to get there? Thanks!

I would approach it like this:

First, the consonant grouping consists of 2 G's and 1 L, and the number of ways they can be arranged is only 3 - this is because the L can come fiirst, or be in the middle, or be at the end: GGL, GLG, or LGG.

Second, the same reasoning applies to the vowels, So there are three ways the vowels can be arranged: EEO, or EOE, or OEE

Fnally, the consonant group can be placed either before all the vowels, or after the first vowel, or after the second vowel, or after the third vowel. That's 4 ways the consonants and vowels can be intermingled. If we let V = vowel and C = consonant, then the four ways are: CCCVVV, VCCCVV, VVCCCV, and VVVCCC.

So the total number of ways to arrange GOOGLE while keeping consonants together is is 3 x 3 x 4 = 36.

Oh, how simple! Thank you. For all of the other questions I practiced that were similar to this, I had no problems but this one really confused me. I had never approached any of them like this. Thank you so much!