10.1 & 10.2
Identify the essentials - How many different words, using all the letters, can be constructed from the letters in the word BINOCULARS?
Order matters, repetition is not allowed..
Why? The order of letters in words need to make sense to make them words and there are no repeating letters in 'BINOCULARS'.
How many permutations of the letters in WATER are there?
120
Why? Use the multiplication counting principle or slot method to find 5! (5 x 4 x 3 x 2 x 1)
Find row 6 of Pascal's Triangle and use it to give expansion of (x+y)6
1x6 + 6x5y1 + 15x4y2 + 20x3y3 + 15x2y4 + 6x1y5 + 1y6
Suppose a coin is tossed 6 times... How many different outcomes are possible?
64 outcomes
Why? 26 = 64
How many ways could you put 8 balls into 5 boxes?
495 ways
Why? C(12,8)
Identify the essentials - The number of different 4-scooped ice cream cones that can be made of 20 flavors.
Order doesn't matter, repetition is allowed.
Why? It does not matter what order the scoops are in and there can be repeating flavors.
How many ways can 3 girls and 2 boys be arranged if the boys need to be next to each other?
48 ways
Why? 3! x 2! x 4
Find the 10th term of (4x+y3)18
12,745,441,280x9y27
Why? n=18 r=10 x=1st part of binomial y=2nd part of binomial... put into form to find term.
If Mr. Leoni tosses a fair coin 10 times, how many ways can he get at least 4 tails?
1,048,565 ways
Why? 410 - (1+10)
How many terms are in the expansion of (x+y+z+x)3?
20 terms
Why? C(6,3)
How many 6-letter strings can be created from the 26 letters of the alphabet if a letter cannot be used more than once?
165,765,600 strings
Why? Use slot method (6 slots) to multiply 26, 25, 24, 23, 22, and 21 together.
A) Evaluate P(16,4)
B) Evaluate P(5,3)
C) Evaluate P(20,7)
A) 43,680
B) 60
C) 390,700,800
Find the 7th term of (2x-3y)10
3,360x4729y6
Why? n=10 r=7 x=1st part of binomial y=2nd part of binomial... put into form to find term.
In how many ways can a team win at least 8 of its first 10 games?
56 ways
Why? 10C8 + 10C9 + 10C10
How many integers from 0 to 9999 have digits that add up to 9?
220 integers
Why? C(12,9)
Among the integers from 1000 to 9999, how many have at least one digit that is a 6 or a 7?
5,416 integers
Why? Multiply integers with 6's and 7's (9000) and integers without 6's and 7's (3584), then subtract them.
Spanish club has 70 members (30 men, 40 women) with an executive board of 3 men and 7 women. How many different executive boards are possible?
75,692,853,600 different exec. boards.
Why? 30C3 x 40C7
Expand (2x-y3)4
16x4 - 32x3y3 + 24x2y6 - 8xy9 + y12
The lacrosse team has won 80% of its games in the past, what is the probability that they will win at least 7 of their first 9 games?
73.82%
Why? n=9 k= 7, 8, or 9 s= 0.8 f= 0.2
A spinner for a game consists of a circle with 5 equal sections of 5 different colors. If an outcome is defined as a certain number of occurrences of each color, how many different outcomes are possible in 10 spins?
1,001 outcomes
Why? C(14,10)
How many different 4 digit numbers can be constructed from the digits 1, 2, 3, 4 if repetition of the digits is allowed?
256
Why? Use slot method to find 4 x 4 x 4 x 4 = 44
There are 5 vowels and 21 consonants in the alphabet. How many different 6 letter "words" (which don't have to make sense) can be formed that contain 2 different vowels and 4 different consonants?
43,092,000 "words"
Why? 21C4 x 5C2 x 6!
21 consonants choose 4, 5 vowels choose 2, 6 letters total (factorial).
Expand (x3-2y)2
x6 - 2x32y + 4y2
Apple gives away new Macbooks by printing the winning code on the back of stickers. If 8% of the stickers produce a winning code, what is the probability that in a pack of 20 stickers that 4 will result in a winning code?
5.23%
Why? n=20 k=4 s= 0.08 f=0.92
In how many ways can 12 new players to a league be added to the numbers currently in 4 teams?
455 ways
Why? C(15,12)