Fundamental Counting Principle
You are going back-to-school shopping. You can choose from 3 binders, 4 highlighters, and 3 colour pens. How many ways can you get one of each?
3*4*3 = 36
How many ways can you arrange 5 people in a row?
5! = 5*4*3*2*1 = 120
How many ways can you choose 2 marbles from a bag of 7 different marbles if you do not put the first marble back?
7* 6 = 7P2 = 42
FCP or perm formula
How many ways can you make a starting basketball line-up of 5 players from a team of 12 players?
12C5 = 792
True or false: order matters when solving a combination counting problem
False, order only matters for permutations
A building has 10 doors. How many ways can you enter and exit the building if you cannot exit from the same door you entered?
10*9=10P2 = 90
___ * ____
enter exit diff
What is 8!/5!
336
How many 4 letter arrangements can be made from the word COMBINE if no letter is used more than once?
7*6*5*4 = 7P4 = 840
A shepherd is asked to choose his 3 favourite sheep to compete in a sheep beauty pageant. How many ways can he choose them from his flock of 10?
10C3 = 120
How many ways can we deal 5 diamonds from just the diamonds in a standard deck of cards?
13C5 = 1287
If a password has a letter followed by 4 digits that cannot be repeated how many different passwords can be made?
26*10 *9*8*7 = 131,040
What is 8!/(2!3!)
3360
There are 45 students are in student council. In how many ways can students be chosen for President, VP, Secretary, and Treasurer?
45P4 = 3,575,880
The general manager of a restaurant chain must select 6 restaurants from 11 for a promotional program. How many possible ways can this selection be made?
11C6 = 462
A shelf can hold 7 trophies. How many ways can the trophies be arranged if there are 10 trophies available?
10P7 = 604800
The code on your car has 5 digits. If the first digit cannot be 0 and it must be even, how many codes are possible? (repeats allowed)
4*10*10*10*10 = 40,000
first # can only be a 2,4,6,or 8 as it can't be 0 or odd
How many ways can you arrange the letters in WINNIPEG?
distinguishable permutation formula:
n!/repeating!
8!/(2!2!) = 10,080
A roofing company has 8 roofing jobs to complete in the next two month. If 5 can be completed in the first month, how many ways can the first 5 jobs be selected?
5*4*3*2*1 = 8P5 = 6720
FCP can be used as don't want repeats or perm form
How many ways can you choose a dance team of 2 ballet dancers and 3 hip-hop dancers from a group of 8 ballet dancers and 7 hip-hop dancers?
8C2 * 7C3 = 28*35 = 980
In how many ways can a foreman, assistant foreman, and secretary be selected from a 12-member group?
12P3 = 1320
A password is 3 letters followed by 3 digits. How many possible passwords have the order consonant, vowel, consonant, even, odd, even? Repetition is possible.
21x5x21x5x5x5 = 275 625
How many vertical arrangements are there of 8 flags if 4 are white, 3 are blue and one is red?
distinguishable permutation formula:
n!/repeating!
8!/(4!3!) = 280
How many 4-digit codes can be made using the digits 1-7, no repetition
7*6*5*4 = 7P4 = 840
FCP or perm formula
Student council has 4 boys and 5 girls. A group of 4 is being chosen to attend a field trip. How many different groups have at least 2 girls going?
4C2 x 5C2 + 4C1 x 5C3 + 4C0 x 5C4
= 60 + 40 + 5 = 105
A test consists of 20 questions, but you are told to answer only 15. In how many different ways can you choose the 15 questions?
20C15 = 15504