Question #a504d

First place can be filled in 10 ways second place in 9 ways third place in 8 ways...etc.

So we have 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 This can be written as 10! ( this is known as 10 factorial).

This idea can be a bit difficult to grasp at first.

These types of problems are called permutations.

The general formula for this is;

#(n!)/(( n - r )!)#

Where #n# is the number of objects and #r# is how many are taken at a time.

So above problem would be:

#(10!)/((10 - 10 )!)# = #(10!)/((0)!)#

Note: #0! = 1#

So we have: #10!# = 3628800

Hope this helps you.

There are #10# different possibilities for the 1 st runner.

For each of these #10# possibilities there are #9# different possibilities for the 2nd place runner. That is there are #10xx9 = 90# possibilities for the first 2 runners.

Similarly for each of these #90# possibilities there are #8# different possibilities for the 3rd place runner. That is there are #10xx9xx8=720# possible combinations for the first 3 runners.

Similarly there are #10xx9xx8xx7# possible combinations for the first 4 runners; and
there are #10xx9xx8xx7xx6# possible combinations for the first 5 runners; and
thethere are #10xx9xx8xx7xx6xx5# possible combinations for the first 6 runners; and
there are #10xx9xx8xx7xx6xx5xx4# possible combinations for the first 7 runners; and
there are #10xx9xx8xx7xx6xx5xx4xx3# possible combinations for the first 8 runners; and
re are #10xx9xx8xx7xx6xx5xx4xx3xx2# possible combinations for the first 9 runners; and
there are #10xx9xx8xx7xx6xx5xx4xx3xx2xx1# possible combinations for all 10 runners.

#10xx9xx8xx7xx6xx5xx4xx3xx2xx1=3,628,800#

Note that this is often written as: #10!# (read "10 factorial").
Most spreadsheets and some calculators have a built-in function for this, so you don't need to do the multiple calculations.