Name:Algebra 2
Date:
Homework: Counting and Pascal’s Triangle
1.Fill in this chart about what happens when flipping a coin 6 times. Hint: The answers you need to evaluate 6C0, 6C1, etc. You can get these values from your calculator.
# of heads / # of tails / # of sequences of this kind6 / 0
5 / 1
4 / 2
3 / 3
2 / 4
1 / 5
0 / 6
2.Fill in this chart about what happens when flipping a coin 8 times.
# of heads / # of tails / # of sequences of this kind8 / 0
7 / 1
6 / 2
5 / 3
4 / 4
3 / 5
2 / 6
1 / 7
0 / 8
3.Suppose a coin is flipped 10 times. How many coin flip sequences are there that contain exactly 3heads?
4.Suppose a coin is flipped 12 times. How many coin flip sequences are there that contain exactly 5 tails?
Conclusion: repeated binomial experiments
A binomial experiment is an action that has two possible outcomes (example: coin flip).
A repeated binomial experiment is a two-outcome action done multiple times.
What we’ve observed involving coin flips actually applies to all binomial experiments.
Here’s the general conclusion:
Suppose that a binomial experiment is repeated n times.
Getting a certain outcome k-out-of-n times can occur in nCk different ways.
Pascal’s Triangle of combination numbers
This triangular number pattern called Pascal’s Triangle is a good way to find combination numbers without having to multiply or use a calculator. Each number in the pattern is the sum of the two numbers above it. For example, 5C2 = 10 comes from adding the 4 and 6 above it.
0C01C0 1C1
2C0 2C1 2C2
3C0 3C1 3C2 3C3
4C0 4C1 4C2 4C3 4C4
5C0 5C1 5C2 5C3 5C4 5C5
6C0 6C1 6C2 6C3 6C4 6C5 6C6 / = / 1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
So now we have three possible ways to get the value of a combination number nCr:
- by writing a fraction and evaluating it
- by using the calculator shortcut underneath the MATH key
- by writing the “Pascal’s Triangle” pattern
Problems
5.Write the next row of the triangles above, then identify the value of 7C3.
6.Write another row of the triangles above, then identify the value of 8C5.
More combination number problems
7.A baseball team has played 7 games. Each game is either a win (W) or a loss (L). Think about all the possible sequences of wins and losses (example: W W L W L L W). Fill in the table below. Hint: Think of wins-and-losses just like heads-and-tails.
7 / 0
6 / 1
5 / 2
4 / 3
3 / 4
2 / 5
1 / 6
0 / 7
8.Suppose that an Italian Sub sandwich can be ordered with or without each of the following items, in any combination:
lettuce, tomato, salt, pepper, oregano
For each number of items listed below, how many different Italian Subs exist?
Write your answers first in nCr form, then as plain numbers.
9.To get the answers to the next problem you will need to make Pascal’s Triangle down to the row that begins “1 10 …” Write out this number triangle in the space here. The beginning of it is shown. Try to figure out the rest without looking back at the previous pages.
1
1 1
1 2 1
10.A pizzeria offers 9 different pizza toppings. How many different pizzas are there with each of the following numbers of toppings?
0 toppings:5 toppings:
1 topping:6 toppings:
2 toppings:7 toppings:
3 toppings:8 toppings:
4 toppings:9 toppings:
11.A competing pizzeria offers 10 different pizza toppings. How many different pizzas are there with each of the following numbers of toppings?
0 toppings:5 toppings:
1 topping:6 toppings:
2 toppings:7 toppings:
3 toppings:8 toppings:
4 toppings:9 toppings:
10 toppings:
Symmetry in the combination numbers
In Pascal’s Triangle of combination numbers, one of the most noticeable features is the symmetry line in the number pattern. Thenumbers in the left half of the triangle match the numbers in the right half. The problems on this page explore this symmetry.
12.Using the method where you write and evaluate fractions, find the values of 13C4 and 13C9. (For 13C9, before multiplying, cancel as many numbers as you can.)
13.Using the method where you write and evaluate fractions, find the values of 11C6 and 11C5.
14.Recall that 13C4 is the number of ways to choose 4 out of a set of 13 objects, and
13C9 is the number of ways to choose 9 out of a set of 13 objects. From this point of view, why does it make sense that 13C4 = 13C9 ?
15.What other combination number is equal to 25C8 ?
(You do not have to find the value. Just answer the question.)
16.In general, what combination number is equal to nCr ?
17.Is there another combination number equal to 10C5 ? Explain.
Adding pattern in the combination numbers
In Pascal’s Triangle of combination numbers, the relationship that enables you to continue the pattern is that each number (except the 1’s on the edges) is the sum of the two numbers above it. Algebraically, this is: nCr = n–1Cr–1 + n–1Cr .
18.Given that 13C4 = 715 and 13C5 = 1287, find the combination number that would appear under these two numbers in Pascal’s Triangle.
19.Which two combination numbers could be added to find 25C8 ?
20.What combination number is equal to 47C15 + 47C16 ?
The next problem is a word problem that seeks to illustrate why the adding pattern arises.
21.There are 10 members on the Portland City Council. The council members need to choose (from among themselves) a 3-person subcommittee.
a.How many different possibilities are there for the members of a 3-person subcommittee?
b.The Mayor of Portland is one of the 10 members of the City Council. How many different possibilities are there for the members of a 3-person subcommittee,
if the Mayor must be included as one of the subcommittee members?
Hint: The subcommittee will consist of the Mayor plus 2 of the other 9.
c.The Mayor of Portland is one of the 10 members of the City Council. How many different possibilities are there for the members of a 3-person subcommittee,
if the Mayor must notbe included as one of the subcommittee members?
d.What combination number addition relationship is illustrated by parts a, b, and c?