Westminister Abbey has 10 bells in its tower. In how many ways can its bells be rung?

Question

"Ringing the changes" is a process where the bells in a tower are rung in all possible permutations.

Answer 1

it's 3,628,800 because....
10!=10x9x8x7x6x5x4x3x2x1=3,628,800

Answer 2

You failed to mention if each ring involves all bells being rung. If it does, and only the order is altered, then the first answer of 10! is correct. If each ring involves a different number of bells being rung, then the answer is somewhat more complicated.

If instead each ring consists of a different set of bells being rung simultaneously, then your question can be restated as how many non-empty sets of bells are there in a set of 10 bells? If this is indeed the question, then the answer of 1023 is correct. A simpler way to obtain the answer is to think of each bell as having a number 1-10, and to then represent a ringing bell with a 1, and a non-ringing bell as a 0. So, for example, the sequence for all the even-numbered bells ringing is 1010 1010 10, ordered from high bell number to low. There are 2^10=1024 possible combinations of this kind. Since we are omitting the one where no bells ring (0000 0000 00), we have a total of 1023 combinations.

Answer 3

You can choose the first bell 10 ways
... you can choose from the remaining 9 bells 9 ways,
... and so on,

resulting in 10*9*8*7...*3*2*1 = 10! ways to permute the bells

Answer 4

10 ways to ring 1 at a time
5 5 ways to ring 2 at a time
120 ways to ring 3 at a time
210 ways to ring 4 at a time
252 ways to ring 5 at a time
210 ways to ring 6 at a time
120 ways to ring 7 at a time
55 ways to ring 8 at a time
10 ways to ring 9 at a time
.. 1 way to ring 10 at a time
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1,023