prove that the sequence of binomial coefficients Cn 0 Cn 1 C

prove that the sequence of binomial coefficients C(n 0), C(n 1), C(n 2),...,C(n n) first increases and then decreases. If n is even the middle one is largest; if n is odd the two middle ones are the largest.

Solution

let f(k)=C(n k)

so f(k)/f(k-1)=C(n k)/C(n k-1)=[n!/{k!*(n-k)!}]/[n!/{(k-1)!*(n-k+1)!}]=(n-k+1)/k k=0,1,.....n

now f(k)/f(k-1)>1 as (n-k+1)/k >1 =>n-k+1>k => k<(n+1) /2   so f(k)>f(k-1)   as k<(n+1) /2

f(k)/f(k-1)=1 as (n-k+1)/k =1 =>n-k+1=k => k=(n+1) /2          so f(k)=f(k-1)   as k=(n+1) /2   and it is an interger.

f(k)/f(k-1)<1 as (n-k+1)/k <1 =>n-k+1<k => k>(n+1) /2           so f(k)<f(k-1)   as k>(n+1) /2

so the coefficients increases for all k such that k<(n+1) /2 then it reaches maximum at k=(n+1) /2 and then decreases for all k such that k> (n+1)/2

when n is even the middle value is (n+1+1)/2 so largest is f(k) where k=(n+2)/2 i.e the middle value

when n is odd the middle values are (n+1)/2 and (n+1)/2+1 so largest are f(k)=f(k-1) for k=(n+1)/2+1 i.e the middle ones. [proved]