How many different bit strings of length 12 contain exactly

How many different bit strings of length 12 contain exactly three ones? at most three ones? at least three ones? an equal number of 0s and 1s? Show that (n k) lessthanorequalto 2^n for all positive integers n and k with 0 lessthanorequalto k lessthanorequalto n.

Solution

6. for all positive integers nCk < = 2^n

example 2C1 = 2; 2C2 =1 ; 2C0 = 1;

if we add ll these we get 4 = 1+2+1 = 2C0 +2C1 +2C2;

3C0 + 3C1 +3C2 + 3C3 = 1+3+3+1 =8 = 2^3 =8;

similrlyy we can applyy for all values of n where sum of the coefficients of any binomial expansion is 2^n;