First slide

Combinations

Question

In an examination a candidate has to pass in each of the papers to be successful. If the total number of ways to fail is 63, how many papers are there in the examination?

Moderate

A

6
B

8
C

14
D

None of these

Solution

Let the number of papers be n.
$∴$ Total number of ways to fail or pass
$^{n} C_{0} +^{n} C_{1} +^{n} C_{2} + \dots +^{n} C_{n} = 2^{n}$ .
But there is only one way to pass, i.e., when he fails in none.
$∴$ Total number of ways to fail = 2ⁿ – 1.
$∴$ From question, 2ⁿ – 1 = 63;
$∴$ 2ⁿ = 64 = 2⁶
$∴$ n = 6.

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