+0

+1
616
8
+477

How many ways are there to divide 12 people into a group of 3, a group of 4, and a group of 5, if Henry has to be in the group of 4?

Sep 6, 2020

#1
0

There are 84,320 ways to divide the 12 people.

Sep 6, 2020
#2
+114522
+2

I am not sure but this is my best guess.

Forget about Henry, there are 11 people which are to be divided into groups of 3,3, and 5

There are 11C5 ways to choose the group of 5

Now there are 6C3 ways to choose a group of 3 BUT this is double counting, I need to divide by 2

so I have    11C5 * (6C3 /2)

Now there are 2 options for where to put Henry.

so I now have    11C5 * (6C3 /2) * 2 = 462*20/2*2 = 9240 option.

Sep 7, 2020
#3
+1

Melody's solution is accurate. It is the same as: 11! /(5! * 3! * 3!) =9,240 ways. Henry can sit with any group of 3.

Sep 8, 2020
#4
+114522
+2

ok, thanks   But I would prefer to describe it as

11! /(5! * 3! * 3!)   divide by 2 because the 3! and 3! are interchangeable. multiply by 2 because there are 2 ways to add in Henry.

=  9,240 ways

I am still having some trouble thinking about it this way though.

This would normally be the answer to a question like this.

How many ways can the 11 letters  aaabbbcccc be arranged in a line.   11!/(3!3!4!)  is the answer.    [I am not going to concern myself with Henry]

For our question it would be how many ways can 11 objects be put in a line if the order of the first 5 does not matte, the order of the next 3 does not matter and the order ot the last 3 does not matter AND the second and third groups of 3 can be swapped around.

NOW it does make sense to me.    11!/(3!3!4!) divided by 2

Thanks guest for making me think about that.

Melody  Sep 9, 2020
edited by Melody  Sep 9, 2020
edited by Melody  Sep 9, 2020
#5
+477
+1

Thank you very much! But 9420 is incorrect

Sep 10, 2020
edited by HelpBot  Sep 10, 2020
#6
+3

You say " Thank you very much! But 9420 is incorrect "!, But the answer given to you is 9240 and NOT 9420 ?!.

Sep 11, 2020
#7
+477
+1

Sep 13, 2020
#8
+114522
0