Question

A determinant is chosen randomly from the set of all determinant of order $4$ with elements $2$ and $3$. The probability that determinant chosen will have an even number of $2^{\prime }$s is

• A. $\frac{2^{15}-2}{16}$
• B. $\frac{2^{15}-2}{2^{16}}$
• C. $\frac{2^8-1}{2^{16}}$
• D. $\frac{2^{15}-1}{2^{16}}$

Answer 1

The correct option is D $\frac{2^{15}-1}{2^{16}}$

Order of given determinant is $4$.
total number of determinants = $2^{16}$
Now number of determinants in which there are even number of $2^{\prime }$s $=$
${}^{16}{C}_0{+}^{16}{C}_2{+}^{16}{C}_4{+}^{16}{C}_6...$ = $2^{15}$ $-1$
Required probability $=$ $\frac{2^{15}-1}{2^{16}}$