05 Permutations and Combinations Notes Filled In.notebook
September 29, 2014
Stats & Discrete Math
Permutations and Combinations
Last week, we figured out the number of possible outcomes for different events by manually multiplying numbers together. This week, we’re going to learn to use two different shortcuts. When we have a group of things, and we’re selecting some or all of them, we can use permutations and combinations. For permutations, _____________________________________________________________. That is, the first person you pick from a group of 10 is considered different than the second person, or the fourth person. This will happen when you’re lining people up, putting pictures on a wall, putting books on a shelf, etc.
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05 Permutations and Combinations Notes Filled In.notebook
September 29, 2014
For combinations, ____________________________________________________________. That is, the first person you pick from a group of 10 is no different than the second or fourth. You use combinations when you’re making groups, teams, or sets, where no one is being assigned a particular role.
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05 Permutations and Combinations Notes Filled In.notebook
September 29, 2014
Permutations
Ex. 1: How many different ways can 5 people stand in a line?
We can do this using our strategies from last week: We can also do this problem using a permutation (since order matters here). The symbol for permutation is _________ where n is the total number of things you have and r is the number you are selecting. If we have 5 people and we’re putting all 5 of them in a line, then we need to enter _________.
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05 Permutations and Combinations Notes Filled In.notebook
September 29, 2014
You can do this on the graphing calculator. • On the TI­nspire, from the calculate screen, hit “menu”, 5:Probability, 2:Permutations. You should then see nPr() on the screen. Enter the two numbers in the parentheses, separated by a comma.
• On the TI­84, first enter the number to the left of the P. Then hit the MATH button, go to the PRB menu and choose nPr. nPr should show up on the screen to the right of the number you already typed in. Type the second number and hit enter. What answer do you get? __________ Does it match what we expected?
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05 Permutations and Combinations Notes Filled In.notebook
September 29, 2014
Ex. 2: You are a museum curator. How many different ways are there for you to hang 13 paintings in your gallery?
Does order matter?
Find the number of ways to hang the 13 paintings:
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05 Permutations and Combinations Notes Filled In.notebook
September 29, 2014
Ex. 3: Your exhibit has been downsized, and you are now only allowed to hang 8 of the 13 paintings. How many different ways are there to do this?
Does order matter?
Find the number of ways to hang the 8 paintings:
Is our answer bigger or smaller than the previous example? Why? 6
05 Permutations and Combinations Notes Filled In.notebook
September 29, 2014
In general, nP1 < nP2 < nP3 < nP4 < nP5, etc.
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05 Permutations and Combinations Notes Filled In.notebook
September 29, 2014
Ex. 4: 20 students are running for senior class officer positions. They can be assigned to president, vice­president, secretary, and treasurer. How many different ways can this happen?
Does order matter?
Find the number of ways to pick the officers:
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05 Permutations and Combinations Notes Filled In.notebook
September 29, 2014
Ex. 5: How many 3­digit numbers are there that contain only the digits 1, 2, 3, and/or 4?
Does order matter?
However, what’s different about this problem compared to the other examples we’ve done today? Find the number of 3­digit numbers:
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05 Permutations and Combinations Notes Filled In.notebook
September 29, 2014
Combinations
Ex. 6: How many groups of 8 paintings can you choose from 13 total paintings?
This is different from our example yesterday in one key way. Here, we are just talking about the groups of paintings and not their order. For situations like this, we use a combination instead of a permutation.
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05 Permutations and Combinations Notes Filled In.notebook
September 29, 2014
On the calculator, the numbers are entered the same way as a permutation with the total first and the number in your group second. •
On the Ti­nspire, go to “menu”, 5:Probability, 3:Combinations, and then proceed as with permutations. •
On the TI­82, enter the first number (n). Then go to MATH, PRB, and nCr is located just below nPr in the PRB menu. Enter the second number just as before, and hit ENTER.
Find the number of groups of 8 paintings:
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05 Permutations and Combinations Notes Filled In.notebook
September 29, 2014
In general, combinations will give you smaller answers than permutations. This makes sense if you think about it for a second. Let’s say you have 3 things, A, B, and C. That would count as just one group (combination), but how many different orders (permutations) could we get?
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05 Permutations and Combinations Notes Filled In.notebook
September 29, 2014
Ex. 7: What if we chose groups of 5 paintings from the 13 total?
Does order matter? Find the number of groups of 5 paintings:
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05 Permutations and Combinations Notes Filled In.notebook
September 29, 2014
Does the answer look familiar? Do you think that’s a coincidence? Do you notice anything about the numbers 13, 8, and 5? 14