A student is allowed to choose atmost (n ) books from a collection of (2 n+1 ) books. If the total number of ways in which he can select atleast one book is 255 , then the value of (n ) is

Step-by-step solution

According to given information, ( aligned & ^ 2 n+1 C₁+ ^ 2 n+1 C₂+ ^ 2 n+1 C₃+ + ^ 2 n+1 C_n=x Let & ^ 2 n+1 C₀+ ^ 2 n+1 C₁+ ^ 2 n+1 C₂+ + ^ 2 n+1 C_n+ & + 2 n+1 2 n+1 =2^ 2 n+1 & 2 x+ ^ 2 n+1 C₀+ 2 n+1 2 n+1 =2^ 2 n+1 & [ ^n C_ n-r = ^n C_r ] & 2 x+1+1=2^ 2 n+1 & [ ^n C₀= ^n C_n=1 ] & x=2^ 2 n -1=255 (given) & 2^ 2 n =256=2^8 & n=4 aligned ) Hence, option (1) is correct.