Solutions to Practice Problems for  the Lander Math Contest
Module 6, Solutions (16-18)

16) a)By definition:
10C6 = 10!/(4!x6!)=7x8x9x10/(2x3x4)=7x3x10=210
    b) Apply the same formula for binomial coefficients:
13C7=13!/(6!x7!)=11x12x13=1716.

17) Let us use the binomial coefficients’ formula and then
simplify the expression:
a)
nCn-2=n!/[2!(n-2)!]= [(n-2)!(n-1)n]/[2!(n-2)!]=n(n-1)/2
b)
nCn-3=n!/[3!(n-3)!]= [(n-3)! (n-2)(n-1)n]/[6 (n-3)!]=n(n-1)(n-2)/6

18) The left side of the formula is n!/[(n-k)!k!]. Then, the right
side would be n!/[(n-(n-k))!(n-k)!]= n!/[ k!(n-k)!].