Combinations(Use paragraph to explain it and give example)?

Question

1.Describe the difference bewtween a combination and permutation. Give examples to illustrate your descriptions.

2.Describle the relationship between 5P3 and 5C3. Then explain how the formula for combinations is related to the formula for permutations.

3.Using the definition of a combination and the values for 8C8 and 8C1, find and describle nCn and nC1 for any whole number n.

Answer 1

1. combination is when arrangements doesn matter. eg ABC and CAB is the same. but for permutation, ABC and CAB are 2 different arrangements.. do you get my point?(:

2.for example you have 5 people, A,B,C,D and E but you have only three seats. 5P3 will give you the number og possible arrangements there are; ADE and EAD are different arrangements. 5C3, on the other hand, will give you how many possible ways are there for 3 people to take the three seats (what i meant is that arrangement does not matter, BDE and DBE are the same)
formula for combinations is n!/[(n-r)!r!] whereas formula for permutations is n!/(n-r)! combinations is where arrangements does not matter, hence it has to be divided by another r!.

3.8C8 = 1, 8C1=8
8C8 is where there are eg. 8 seats for 8 people. since arrangement does not matter in combinations, there is only 1 possible way.
8C1 is where there are 8 people but only 1 seat. hence there can be 8 different ways for the 8 people to occupy the only place(:

hope u understand more now!(:

Answer 2

Combination refers to the number of groups one could select from a larger set

e.g ways you select 3 boys from a group of 5

this is symbolized as 5C3

if you want to know how these 3 boys could be seated on three seats for example, that is the number of permutations. 3 people could be arranged in 6 ways = 3 x 2 x 1

if you take the order also to account when you selected the combinations above there should be 5C3 x 6 ways of doing it . THis is symbolized as 5P3

So for any n and r whole numbers nPr = 1x 2 x 3 x 4 x ....xr X nCr ------------------(A)

1x 2 x 3 x .... r is called factorial r and written as r!

It is easy to find the number of ways n objects could be arranged ( = n ! = 1 x 2 x 3 x n ) you can think of seating n persons on n seats
take the first person he has n seats at his disposal so after he takes a seat then the next one has (n-1) ways of sitting so for the first two there are n x (n-1) ways

for the nexr one there are (n-2) seats yto choose from , next has (n-3) and the last has only one seat left.

So the number of ways is n! ( n x (n-1) x (n-2) X ... X 1 )

now lets take permutations of arranging three persons in n seats

from the above argument it should be clear that there are n x (n-1) x (n-2) ways

that is n P 3 = n x (n-1) x (n-2)

now in factorial language it could be written as = n! / (n-3) !

So for number of ways for r people on n chairs is n P r = n! /(n-r)!


now consider the equation A above which states n Pr = r! x nCr

therefore we can write nCr = n!/((n-r)!r!)

by definition one takes 0! = 1 (remember 1! is also 1)

so 8C8 should be = 1 because there is only one way to choose 8 objects from a group of 8 objects (i.e. take all of them)

8C1 should be equal to 8 since from 8 objects one can choose 1 item in 8 ways i.e. take each one at a time

you could use the formula too 8C8 = 8!/8! 0! =1 (remember 0! =1)

8C1 = 8!/7! 1 ! = 8

useful formulas

nC1 = n , nC0 = 1 ,

nCr = n C (n-r) this can be verified from formula as well as logical thinking, if one selects r objects from a group of n that automatically deselects n-r objects ( or selects ! mathematics is does not know whether we are in the act of selection or rejection)
so ther must be exactly the same number of ways to select n-r objects from n objects = n C (n-r)

THer are other beautiful equations you could think of each of these could be proven algebraically or logically.


HAVE FUN !

PLease bear with me for using "HE" for gender neutral nouns. I diid it to avoid overuse of my rickety keyboard!

Answer 3

first remember this

combination is SELECTION

permutation is ARRANGEMENT


consider nCr this gives the no of ways in which i can select 'r' things out of 'n' things

while nPr gives the ways in which you can arrange 'r' things in
'n' places

here is the direct formula

nPr = n! / (n-r)!

(n! = n * n-1 * n-2 * . . . * 3 * 2 * 1 )

nCr = n! / { (n-r)! r!}


from the formulae u can see that

nPr = nCr * r!


consider any no. n

nCn = n! / { n! * (n-n)! }

= n! / (n! * 0!)

note that 0! =1
thus
nCn = n! / n! = 1


now nC1 = n! / ( 1! * (n-1)! )

from the definition of factorial u can see that

n! = n * n-1 * n-2 * . . . * 3 * 2 * 1

= n * ( n-1 * n-2 * . . . * 3 * 2 * 1 )

= n * (n-1)!

using this in above eqn

nC1 = ( n * (n-1)! ) / (n-1)!

= n