Page 180 - DMTH202_BASIC_MATHEMATICS_II

Unit 13: Combinations

Notes
52!
 * Expand 52! until it obtains to 48! which is the larger! in the den.
48!4!
52.51.50.49.48!
 * Cancel out 48!’s
48!4!

52.51.50.49

4.3.2.1
= 270725
If you contain a factorial key, you can place it in as 52!, divided by 48!, divided by 4! and then
press enter or =.

If you don’t contain a factorial key, you can abridge it as revealed above and then enter it in. It
is most likely best to simplify it first, since in some cases the numbers can get pretty large, and
it would be awkward to multiply all those numbers one by one.
This signifies there are 270,725 different 4 card hands.

Example: 3 marbles are taken at random from a bag comprising 3 red and 5 white marbles.

Solve the following questions (a – d):
(a) How many different draws are there?
This would be a combination problem, since a draw would be a collection of
marbles without considering order. It is like taking handful of marbles and gazing at
them.
Observe that there are no particular conditions positioned on the marbles that we draw, so
this is a simple combination problem.
If we were putting these marbles in any sort of order, then we would require using
permutations to solve the problem.
But here, order is not an issue, so we are going to use combinations.
First we need to find n and r:
If n is the number of marbles we have to select from, what do you think n here?
If you said n = 8 you are right!!! There are 3 red and 5 white marbles for a total of 8
marbles.
If r is the number of marbles we are taking at a time, what do you think r is?
If you said r = 3, pat yourself on the back!! 3 marbles are taken at a time.
Let us place those values into the combination formula and observe what we obtain:
! n
r
n
C  * = 8, = 3
n r

r
(n r )! !
8!
C  *Eval. inside ( )
8 3
!
)

( 8 3 3!
8! *Expand 8! until it obtains 5! which is the larger ! in the den.
5 3!
!
*Cancel out 5!'s
.
.
8 7 6 5!
.

!
5 3!
.
.
8 7 6

3 2 1
.
.
 56
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