1. ## Another Permutation question

I'm having trouble with the last part of this question:

A combination lock has 60 numbers on it, from 0 to 59
A) how many 3 number combinations are possible
60^3 = 216000
b) How many three number combinations are there that do not repeat any numbers in the combination
60nPr3 = 205 320
c) using parts a and b determine how many three number combinations have at least two numbers the same
This is the question that I do not understand how to get, here is what I did:
60nPr2 = 3540 60^2 = 3600
I added these numbers together and got
7140

2. Originally Posted by tmas
I'm having trouble with the last part of this question:

A combination lock has 60 numbers on it, from 0 to 59
A) how many 3 number combinations are possible
60^3 = 216000
b) How many three number combinations are there that do not repeat any numbers in the combination
60nPr3 = 205 320
c) using parts a and b determine how many three number combinations have at least two numbers the same
This is the question that I do not understand how to get, here is what I did:
60nPr2 = 3540 60^2 = 3600
I added these numbers together and got
7140
They gave you a hint that is to use both parts a and b.

You know the total number of possible lock combinations 216000

You also know that 205320 of do not repeat any number.

So all that is left are combinations that has two of the same numbers or all three numbers the same, lets call this x then

$\displaystyle 205320+x=216000$ this gives the solution

3. Hello, tmas!

A combination lock has 60 numbers on it, from 0 to 59.

(a) How many 3-number combinations are possible?
. . $\displaystyle 60^3 \,=\, 216,\!000$ . Right!

(b) How many 3-number combinations are there
. . .that do not repeat any numbers in the combination?
. . . . . $\displaystyle _{60}P_3 \,=\, 205,\!320$ . Right!

(c) Using parts a and b, determine how many 3-number combinations
. . .have at least two numbers the same.

This is the question that I do not understand how to get.

Here is what I did:
. . $\displaystyle _{60}P_2 \,=\, 3540,\;\; 60^2 \,=\, 3600$

I added these numbers together and got $\displaystyle 7140.$

This is the wrong answer though . . . The correct answer is 10,680.

This answer happenes to be the difference of the answers in (a) and (b).
Why is this so?

In part (a) you found the total number of possible combinations.

In part (b) you found the number of combinations with no repeated numbers.

Hence, the difference is the number of combinations with some repeated numbers.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

A random thought . . .

On a combination lock, the order of the three numbers is important, isn't it?

Then why don't they call them "permutation locks"?