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# Probability and Permutations

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Practice Probability and Permutations
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## Locker Combinations and Permutations

### Topic

Locker Combinations and Permutations

• Permutation
• Combination

### Student Exploration

#### How many different locker combos can a combination lock have?

It is Fall and school is back in session, Denise needs to buy a lock for her school locker. She wants her belongings to be safe, but she doesn’t want to carry around a key and would rather have a lock with a combination. Before going shopping she decides to research locks online and wants the deciding factor to be the number of locker combination codes that the lock has, that way a thief will have a nearly impossible task of breaking her lock’s combination code.

Research different locks online and find three different models with different designs and set-ups for the locker combos. Then determine which one you would recommend to Denise as the safest for protecting her belongings, using your math work as evidence!

You can calculate the total number of locker combinations with $10 \cdot 10 \cdot 10=10^3=1000$.

### Extension Investigation

One of the locks Denise considers is a circular lock with 20 numbers. To open it she turns the dial to the first number, then she turns the dial to the second number, and then she turns it to the third number. She may not repeat numbers in her locker combo.

a. Why does this situation represent a Permutation?
This situation is a Permutation because the order of the three numbers makes a difference in the locker combination code. For example the code 10, 15, 8 is a different code than 8, 10, 15.
b. Write an expression for the number of different combination codes that are possible with this lock?
$_{20}P_{3} \ or \ 20 \cdot 19 \cdot 18$ are examples of expressions for this lock’s combination code.
c. If Denise could use four numbers in her locker combo, how would you approach this task?
In the $_nP_r$ expression $r$ would equal 4 instead of 3. Or, if you used the counting principle then you would multiply four numbers instead of three, such as $20 \cdot 19 \cdot 18 \cdot 17$.
d. If Denise could repeat numbers in her locker combo, how would you approach this task?
Then you would need to use the permutations with repetitions formula or when using the counting principle it would be $20 \cdot 20 \cdot 20 = 20^3$. For more info on calculating permutations with repetitions visit, http://www.ck12.org/concept/Permutations-with-Repetition/.''
e. If Denise could choose from 40 different number, rather than 20, how would you approach this task?
In the $_nP_r$ expression $n$ would equal 40 instead of 20. Or, if you used the counting principle then you would start your multiplication expression with 40, such as $40 \cdot 39 \cdot 38$.
f. What would need to change in the problem in order for it to represent a Combination?
In order for this to be a Combination problem, the order of the numbers in the combination lock wouldn’t matter. If you put the code 10, 15, 8 in, it would be the same locker code as 8, 10, 15.
g. Why doesn’t it make sense for locks to be “Combination Locks”?
It doesn’t make sense because it isn’t a Combination, it is a Permutation. They should be called “Permutation Locks.”

### Connections to other CK-12 Subject Areas

• Combination Problems
• Permutations and Combinations Compared
• Permutations with Repetition.