9.5 Probabilities of Various Combinations

This section discusses another way to calculate probability of specific situations. Also, there are some new (easier) notations, and some calculator functions! If you want, you can call this section “Permutations VS Combinations,” as Mr. French has listed it for Friday’s quiz topics list. Being able to distinguish the combination from the permutation is pretty important, and probably will be for this next quiz!

Terminology:
1. Permutations: Looking @ the # of possible outcomes in a situation when order is important (as we learned from Nick yesterday).

2. Combinations: Looking @ the # of possible outcomes in a situation when order is not important.

A number of combinations = total number of permutations / # of permutations of each one combination

So, when we are asked on the quiz: according to their definitions, what is the conceptual difference between a permutation and a combination? We’ll nail it! ☺ Significance of order is what counts!!

Notation:
1. Permutations: nPr = n!/(n-r)!
2. Combinations: nCr = n!/(n-r)!r!
• n = total
• r = amount being chosen
• (n-r) = amount not being chosen, a.k.a. amount left

Examples:
Permutations: You’re hosting the after party for a school dance, say, last Friday’s Toga Dance. You’ve got a huge couch that can seat 10 people- no sitting on laps allowed, according to your parents, since heaping on more than maximum carrying capacity will dent the seat pillows beyond repair. Anyway, fifty people show up. They all want to sit on the couch. How many possible ways could you seat the guests?

n = total = 50
r = amount being chosen = 10
So…
50P10 = 50!/(50-10)! BUT WAIT!

Nick taught us yesterday how to do individual factorials by calculator…but we can also do the entire permutations with one function, so lets do that.

• Enter n, which is 50.
• Click MATH, then go to PRB (all the way to the right)
• Scroll down to & choose the nPr option
• Enter r, which is 10.

DANG! The answer is 37,276,043,020,000,000.

Combinations: Same situation, but this time, you want to know how many combinations you can make. The order of the people don’t matter, you just can’t make the same combination more than once.

n = total = 50
r = amount being chosen = 10
So…
50C10 = 50!/(50-10)!10! BUT WAIT! AGAIN!

Like the permutation calculator shortcut, there’s also a combination shortcut. The only thing you need to do differently is that when you scroll down in the PRB section, choose the nCr option.

The answer will be 10,272,278,170 – only.

• Keep in mind that there will be less combinations than permutations. Because order matters, several permutations could be different arrangement of the same combination.

Concept Examples:
Permutations: Write 8P3 as ratio of factorials. Write the answer in terms of the number of elements of the set & number of elements selected.
Answer: 8!/5!
• 8P3 = 8!/(8-3)! = 8!/5!

Combinations: Write 6C2 as a ratio of a factorial to a product of factorials.
Answer: 6!/4!2!
• 6C2 = 6!/(6-2)!2! = 6!/4!2!

Combinations in terms of Permutations: This time, write 6C2 as a ratio of permutations
Answer: 6P2 / 2P2
• Looking back to the Terminology section:
A number of combinations = total number of permutations/# of permutations of each one combination

So… 6C2 = 6P2 / 2P2

That’s it guys! If anybody needs any extra help, here’s a good, simple site:
http://www.themathpage.com/aPreCalc/permutations-combinations.htm

And for those of you that like really convoluted explanations with pretty pictures, here’s another good one, just in case (hey, additional sample problems!):
http://www.mathagonyaunt.co.uk/STATISTICS/ESP/Perms_combs.html

MADDIE’S NEXT! Ah… barely missed the weekend.

Prep Enthusiasm
I’d like to dedicate this personalization to one of our Prep geniuses: DR. PARKER! (Sorry, Mr. French. We all know of your genius, but you already have an entire interview in one of the calculus blogs!!) Dr. Parker is a crazy composer, as in CRAZY GOOD! Sometimes, the instrumental music dept. gets the opportunity to play some of his works for plays, choral & orchestral, etc. But what I really wanted to get out there was that he has a website with tons of recordings that you can download or listen to for free. It’s www.robertparkermusic.com. It’s really quite an impressive collection in terms of both quantity & quality.