Anwar is stringing wooden beads on a leather thong. He has twelve beads of various sizes and will tie a knot in the thong when he has put them all onto it. How many different necklaces could he make? (Hint: Does the exact opposite order of beads produce a different necklace?)
Its 12!/2 man
12!/2
Well, Anwar must be really "stringing" it along with those wooden beads! Now, let's figure out how many different necklaces he can make.
Since Anwar has twelve beads of various sizes, we first need to figure out how many ways we can arrange those beads on the leather thong. This is where we can let our imagination "knot" run wild!
To calculate the number of combinations, we can use the formula for permutations, which is n! (n factorial). In this case, n represents the number of beads. So, n = 12.
However, we also need to consider the hint. Does the exact opposite order of beads produce a different necklace?
Well, let's picture it this way: imagine if Anwar had threaded the beads starting from the opposite end of the leather thong. Would it look any different? I don't think so! It would just be the same necklace flipped around.
Therefore, we have to divide our total number of combinations by 2. This is because each necklace can be flipped, resulting in an identical necklace.
So, the number of different necklaces Anwar can make is (12! / 2). Now, I won't calculate that for you because it would take forever, and we have better things to do, like telling even more jokes!
But don't worry, Anwar has plenty of options to "beadazzle" himself with different necklaces. Just be sure to compliment his fashion sense once you see the final result!
To find the number of different necklaces Anwar can make, we need to consider whether the exact opposite order of beads produces a different necklace.
Let's break this problem down step by step:
Step 1: Find the number of different arrangements of the twelve beads without considering the opposite order.
Since there are twelve beads, the number of different arrangements can be calculated using the factorial of twelve: 12!.
Step 2: Account for the opposite order of beads.
If the opposite order of beads does not produce a different necklace, we need to divide the number of different arrangements by 2. This is because each arrangement will have a corresponding arrangement with the beads in reverse order.
Therefore, the number of different necklaces Anwar can make is given by:
Number of different necklaces = (Number of different arrangements) / 2
Substituting in the value of the number of different arrangements:
Number of different necklaces = 12! / 2
Now, we can calculate this value:
12! = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
Dividing this value by 2 gives us the final answer:
Number of different necklaces = (12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / 2
Calculating this expression will give us the total number of different necklaces Anwar can make.
This type of problem always leads to a lively discussion.
Problem: can the beads be moved past the claps ?
If so, does moving one or more beads from one side to the other without opening the clasp constitute a different necklace?
The general answer to that is no.
(Think of some keys on a keyring.
Flipping the keys around on the ring does not produce a different keyring arrangement, nor does turning the keyring over, only opening the ring itself will produce a new keychain. )
So here is my solution:
Use one of the beads as a "marker", then the others can be arranged in 11! ways.
We then have to divide by 2 to eliminate the opposite order of beads which would not be a new necklace
so I would say: 11!/2