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VITEEE 2012 Paper

Section: Mathematics

Question : 110 of 120

Marks: +1, -0

There are 5 letters and 5 different envelopes. The number of ways in which all the letters can be put in wrong envelope, is

Validate

Solution:

Required numbers

= 5!

\left[ \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} \right]

= 44

Note that if r (0 ≤ r ≤ n) objects occupy the original places and none of the remaining (n - r) objects occupies its original places then the number of such arrangements =

\,{}^{n}C_r (n-r)!

[ 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!}

+ ... +

(-1)^{n-2} \frac{1}{(n-r)!} ]

5 letters 5 directed envelopeAll the of letters Put wrong wrong :

Number of ways

=

= \left[ 1 - \; \frac{1}{1!} + \; \frac{1}{2!} - \; \frac{1}{3!} + \; \frac{1}{4!} - \; \frac{1}{5!} \right] 5!

= \left[ 1 - 1 + \; \frac{1}{2} - \; \frac{1}{6} + \; \frac{1}{24} - \; \frac{1}{120} \right] 5!

= \left[ \; \frac{2}{6} + \; \frac{4}{120} \right] 5!

= \left[ \; \frac{1}{3} + \; \frac{1}{30} \right] 5!

= \left[ \; \frac{11}{30} \right] 120

= 44

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