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 $2 ’s$ 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

Answer: (D)

Order of given determinant is $4$.
$\therefore $ Total number of determinants $= 2^{16}$
Now, number of determinants in which there are even number of $2’s$
$= \,{}^{16}C_2 +\,{}^{16}C_4 + ... + \,{}^{16}C_{16}$
$ = ^{16}C_0 + \,{}^{16}C_2 + ... + \,{}^{16}C_{16} - \,{}^{16}C_{0} = 2^{15} - 1$
$\therefore $ Required probability $ = \frac{2^{15} - 1}{2^{16}}$