If m < n are positive integers prove that nn/(mm (n-m)n-m) > n!/( m! (n-m)! ) > nn/( mm(n+1) (n-m)n-m).
Solution
The key is to consider the binomial expansion (m + n-m)n. This is a sum of positive terms, one of which is nCm mm(n-m)n-m, where nCm is the binomial coefficient n!/( m! (n-m)! ). Hence nCm mm(n-m)n-m < nn, which is one of the required inequalities.
We will show that nCm mm(n-m)n-m is the largest term in the binomial expansion. It then follows that (n+1) nCm mm(n-m)n-m > nn, which is the other required inequality.
Comparing the rth term nCr mr(n-m)n-r with the r+1th term nCr+1 mr+1(n-m)n-r-1 we see that the rth term is strictly larger for r ≥ m and smaller for r < m. Hence the mth term is larger than the succeeding terms and also larger than the preceding terms.
© John Scholes
jscholes@kalva.demon.co.uk
11 Apr 2002