In an examination, a student has to write exams in 8 different subjects. He is declared fail if he fails in…
In an examination, a student has to write exams in 8 different subjects. He is declared fail if he fails in at least one subject. The number of ways in which he can fail is
127
256
255
7
Solution
The number of ways, in which student can fail
$=$ Number of ways, he can (fail in 1 exam + fail in
2 exams $+\ldots . . .+$ fail in 8 exams)
$\begin{aligned} & ={ }^8 \mathcal{C}_1+{ }^8 \mathcal{C}_2+{ }^8 \mathcal{C}_3+\ldots+{ }^8 \mathcal{C}_8 \\ & =2^8-{ }^8 \mathcal{C}_0 \\ & \quad\left[\because{ }^n C_0+{ }^n C_1+{ }^n C_n=2^n\right] \\ & =256-1=255\end{aligned}$