Q56 / 160 · AP EAMCET 2021 (25 Aug Shift 2) · Mathematics

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

  • A
    127
  • B
    256
  • Correct answer
    C
    255
  • D
    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) $\begin{aligned} & ={ }^8 \mathcal{C}_1+{ }^8 \mathcal{C}_2+{ }^8 \mathcal{C}_3+\ldots+{ }^8 \mathcal{C}_8 \\ & =2^8-{ }^8 \mathcal{C}_0 \\ & \quad\left[\because{ }^n C_0+{ }^n C_1+{ }^n C_n=2^n\right] \\ & =256-1=255\end{aligned}$

Hence, the correct answer is option: 'C'.

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