11.3 Combinations

When items are selected from the same group

No item is used more than onceThe order of items makes no difference (this bullet differentiates combination from permutation)

When do we use combination?

20 students are running for 3 positions: president, secretary and treasurer. The student with the highest number of votes becomes the president, the second highest becomes the secretary and then followed by the treasurer. How many different outcomes are possible?

5 out of 70 faculty are needed to form a discipline committee. How many outcomes are possible?

How many ways can you select 5 movies from a list of 100 great movies?

In a car race where 15 cars are entered, how many ways can the first three finishers come in?

Permutation or Combination?

The notation nPr means the number of permutations of n things taken r at a time

The notation nCr means the number of combinations of n things taken r at a time

Permutation and Combination

Sample: Find the number of permutation of the letters ABCD taken three at a time.nPr = 4 P 3 = 24

ABC ABD ACD BCDACB ADB ADC BDC

BAC BDA CDA CDBCAB DAB DAC DBCCBA DBA DCA DCB

Even though there are 24 permutations, there are only 4 combinations (ABC, ABD, ACD, BCD)

Permutation and Combination cont.

Since nCr = nPr r!

nCr = ___n!___ (n – r)! r!

Or just go to MATH, PRB and enter number 3:nCr

Formula for Combination

Example 15 out of 70 faculty are needed to form a discipline committee. How many outcomes are possible?

n C r = 70 C 5 =

12, 103, 014

Example 2In a game of poker, a

player is given 5 random cards from a standard 52-card deck. How many different combinations are possible?

52 nCr 5

= 2,598,960

Example 3The minimum

jackpot in Maryland Mega Million Lottery is $12 million. To play, a person needs to select 5 numbers from 1-56 and select 1 number from 1-46. How many combinations are possible?

175,711,536 combinations

Example 4The US senate of the

111th Congress consists of 59 Democrats and 41 Republicans. How many committees can be formed if each committee must have 3 Democrats and 2 Republicans?

26,657,380

Class work: p. 578, #s 1-5, 11-15, 27-29, 37, 38

Homework: p. 578, #s 6-10, 16-26, 30-36, 39

Quiz next meeting (11.4 Probability and 11.3 Combinations)

Assignment