Can Anyone explain the Difference between Perms and Combinations?

Question

I am not sure the book doesn't really help... pertaining to the equations nPr and nCr, I need help knowing when to use which equation.

The questions I am answering are
Express each of the following expressions in the form nPr or nCr.
a)27x26x25x24/ 4 x 3 x 2 x 1 For whish I have ..../4!
b) 30 x 29 x 28 x 27 x 26

All I want to know is how to tell to use each form, not the answers. Thanks!

Answer 1

permutation implies a particular order/pattern or arrangement is required but a combination implies no order/pattern or arrangement is required.
so when you read the problem then read into it to see if their is any requisite of pattern.
I will give an example: a deck of cards consists of 52 cards all different.
A hand is a set of 5 cards but all should be in a particular pattern as decided by the players while setting the game rules, for eg one such setting can be all 5 in a hand should be either aces or spades or clubs or diamonds.
1. so when the question says, find the number of ways a hand can be drawn out of a deck it means a counting 5 cards in a single hand but in ordered way, so it means per or 52P5
2. but if the question says how many hands can be drawn out of a deck , then the hands [ though each one individually be in orderneed by implicit definition] need not be in order. and here we are not counting the number of ways for a single orderd hand but are counting number of total such hands possible. so here we go for combination, 52C5. The formula takes care of implicit orderliness in each hand automatically.
i hope now it is clear.

Answer 2

Permutations involve order, combinations do not.

Notice that

nCk = nPk/k! that's because the k! divides out the ordering of the k objects.

a) You make 4 choices, without replacement, from 27 objects. The denominator is the ways that you can arrange these 4 choices. If you don't care about order then you'd use this

this is 27C4
think of a poker hand, it's 52C5, choosing 5 cards from 52 without replacement, and without regard to order, since you don't care what order the cards are dealt.


b) You're choosing 5 objects from 30 without replacement, ordering counts. This is a permutation.

These are the two basics of combinatorics and is used to develop almost all discrete probability functions.

Answer 3

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Answer 4

nPr = n!/r!
nCr = n!/[r!(n-r)!]

so for example,

6P2 = 6! / 2! = 6x5x4x3x2x1/2x1 = 6x5x4x3

but

6C2 = 6! / (2!4!) = 6x5x4x3x2x1/2x1x4x3x2x1 = 6x5/2x1

Does that help?

Answer 5

per is used when the arrangement of objects matter
eg, a pssword
com is used when the arrangement doesn't matter
eg, selecting a soccer team

Answer 6

27C4=27!/4!*(27-4)!
30P5=30!/(30-5)!