PROBABILI

TY

FACTORIALS, PERMUTATIONS AND COMBINATIONS

nPr=Pn,r =

(Read: Permutation of n choose r.)

Example: 7P3=

FACTORIALS, PERMUTATIONS AND COMBINATIONS

Combinations

nCr= Cn,r =() =

7C3=

Answer : 35

The sophomore class needs to select a committee of 10 out of all of its members to help decorate for Homecoming. The sophomore class has 97 members.

The junior class must elect five officers (pres, VP, treas, sec, historian) for next year. There are 98 juniors in the class.

If there are 6 seniors on a team. How many ways can you arrange those seniors for a group picture?

PERMUTATIONS AND COMBINATIONS

If order matters, use permutations.If order does not matter, use combinations.

Your calculator can find these answers for you.Hit the MATH button. Then scroll across to PRB.#2 nPr is for permutation and #3 nCr is for combinations. To use you must type in the n# first. For instance, 97C10, should be typed in 97 nCr 10 then hit enter. Answer from calculator: 1.257646973 E 13 

A hand of 7 different cards are dealt from a deck of 52 cards.

a) How many different hands are possible?

b) How many hands with 3 spades are possible?

c) How many different hands with 3 spades and 2 diamonds are possible?

How many 5-card hands with 3 red cards and 2 black cards can be dealt from a deck of 52 cards?

There is a soccer team of 17 boys, 5 of whom are seniors. If two captains are chosen and one is a senior and the other one is not, how many ways can you choose captains?

Four balls are selected from an urn that contains 7 red balls and 3 black balls. How many different selections containing 3 red balls and 1 black ball are possible?

A sample of 8 items is selected from a lot of 100 items known to contain 10 defective and 90 nondefective items. How many different samples with 3 defective and 5 nondefective items are possible?

 How many dancing pairs are there for a group of 3 boys and 2 girls?

How many four digit integers can be formed using the digits 1,2,3,4,5,6 (repetitions allowed)?

How many four digit integers can be formed using the digits 0,1,2,3,4,5, (repetitions allowed)?

How many four digit even integers can be formed using the digits 0,1,2,3,4,5 (repetitions allowed)?

 

How many five digit integers can be formed using the digits 0,1,2,3,4,5 (without repetition allowed)?

 

In how many ways can the letters R, S, T, and U be arranged without repetition?

Counting Principle

If a set A contains r elements, set B contains s elements, and A and B share t elements, then the total of elements between the two sets is r + s - t.

B contains s elements

t elements

A contains r elements

A B

Counting Principle

Example with two sets:

A certain kind of item is considered defective if it has a major defect, a minor defect or both. In a batch of 25 defective items, 20 have major defects and 14 have minor defects. How many items in the batch have both major and minor defects?

Counting PrincipleExample with three sets:

In a pollution study of 1500 US rivers, the following data were reported:

520 were polluted by sulfur compounds

335 were polluted by phosphates

425 were polluted by crude oil

100 were polluted by crude oil and sulfur compounds

180 were polluted by sulfur compounds and phosphates

150 were polluted by phosphates and crude oil

28 were polluted by sulfur compounds, phosphates and crude oil

 How many of the rivers were polluted only by crude oil?

 How many of the rivers were polluted by at least one of the three impurities?

 How many were not polluted?