Find the number of permutations of 10 things taken 7 at a time.

10 P 7. plug into you calculator to solve or use the formula:  

n P r = n! / (n - r)!  

so 10! / 3! = 10x9x8x7x6x5x4x3x2x1 / 3x2x1 = 10x9x8x7x6x5x4 = 604,800.

what

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4