First slide

Combinations

Question

Aiay writes letters to his five friends and addresses the corresponding. The number of ways can the letters be placed in the
envelops so that atleast two of them are in the wrong envelopes are

Moderate

A

120
B

125
C

119
D

124

Solution

Required number of ways $= \sum_{r = 2}^{5}^{5} C_{5 - r} D (r)$
$\begin{array}{l} = \sum_{r = 2}^{5} \frac{5!}{r! (5 - r)!} \cdot r! \{1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \dots + \frac{(- 1)^{r}}{r!}\} \\ = \sum_{r = 2}^{5} \frac{5!}{(5 - r)!} \{\frac{1}{2!} - \frac{1}{3!} + \dots + \frac{(- 1)^{r}}{r!}\} \\ = \frac{5!}{3!} \{\frac{1}{2!}\} + \frac{5!}{2!} \{\frac{1}{2!} - \frac{1}{3!}\} + \frac{5!}{1!} \{\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!}\} + \frac{5!}{0!} \{\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!}\} \\ = 10 + 20 + (60 - 20 + 5) + (60 - 20 + 5 - 1) \\ = 10 + 20 + 45 + 44 = 119 \end{array}$

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