Pretty Easy Placement....

Probability Level 3

There are three letters and three corresponding envelopes. The letters are placed in envelopes at random. Find the probability that none of the letters is placed in the right envelope.

The answer is in the form of a b \frac{a}{b} so enter it as a + b .

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The answer is 4.

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Harshvardhan Mehta by Harshvardhan Mehta , 23, India

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2 solutions

Number of ways of placing three letters in three envelopes = 3! = 6 ways

So. P(All letters are wrongly placed) = 2 6 \frac{2}{6} = 1 3 \frac{1}{3} = a b \frac{a}{b}

⇒ a + b = 1 + 3 = 4 \Rightarrow a + b = 1 + 3 = \boxed{4}

L1 L2 L3
E2 E3 E1

OR

L1 L2 L3
E3 E1 E2
3 Helpful 0 Interesting 0 Brilliant 0 Confused

First of all there are 3 envelopes → 3 ! \rightarrow 3! ways of arrangement .

The answer corresponds to number of Derangements of 3 objects 3 ! ⋅ ( 1 + ∑ n = 1 3 ( − 1 ) n 1 n ! ) = 3 ! − 3 ! + 3 − 1 = 2 3!\cdot (1 + \sum_{n=1}^{3} (-1)^{n}\dfrac{1}{n!} ) = 3!-3!+3-1=2

The answer is 2 6 = 1 3 → 4 \frac{2}{6} = \frac{1}{3} \rightarrow 4

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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