Having some trouble with Anagrams and Combinatorics?

[PinkProfessor90]

Hi, everybody! Any help would be greatly appreciated regarding Combinatorics. This is Chapter 6 from Manhattan Prep's Word Problems book.

I cannot seem to understand why we divide by two factorials. Here's the example from the book:

"If 3 of 7 standby passengers are selected for a flight, how many different combinations of standby passengers can be selected?"

In this problem, we are supposed to divide by two factorials, but in the next example, we only divide by one:

"If 7 people board an airport shuttle with only 3 available seats, how many different seating arrangements are possible?"

Is there an easy way to determine when we have to divide by two factorials?

Thank you so much!

[tommywallach]

Hey Pink Professor,

I hate the anagram grid. Allow me to suggest the Slots method:

1) Find # of slots (how many things do you get to pick)
2) Fill in each slot (how many choices do you have for each slot)
3) Multiply all slots together. If there are any slots where order doesn't matter, divide by the # of slots factorial.

TADA!

"If 3 of 7 standby passengers are selected for a flight, how many different combinations of standby passengers can be selected?"

_ _ _

7 * 6 * 5 = 210, but order doesn't matter (if Steve, Dave, and Sandy are selected, that's the same as Dave, Steve, and Sandy being selected), so divide by 3! = 35.

"If 7 people board an airport shuttle with only 3 available seats, how many different seating arrangements are possible?"

7 * 6 * 5 = 210. Order matters in this case, because we are arranging seats (i.e. Steve - Dave - Sandy is different from Sandy - Dave - Steve)

Hope that helps (and if you don't like my method, the answer to your original question is simply because order matters in the second example, and doesn't matter in the first example)!

-t

[PinkProfessor90]

Thank you so much! :)

[tommywallach]

Glad to help!

-t