JAMIA 2023 — Mathematics PYQ

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Question

JAMIA 2023 — Mathematics PYQ

JAMIA | Mathematics | 2023

A person writes letter to 6 friends and addresses the corresponding envelopes. Let’x’ be the number of ways so that at least 2 of letters are in wrong envelopes and ‘y’ be the number of ways so that all letters are in wrong envelopes.Then x- y =?

Choose the correct answer:

Explanation

Step 1: Find $x$ (At least 2 letters in wrong envelopes)

The total number of ways to put 6 letters into 6 envelopes is $6!$ .

The condition "at least 2 letters are in wrong envelopes" is the same as:

\text{Total ways} - (\text{Ways where 0 letters are wrong}) - (\text{Ways where exactly 1 letter is wrong})

Total ways: $6! = 720$
0 letters wrong: This only happens in 1 way (all are correct).
Exactly 1 letter wrong: This is impossible ( $0$ ways) because if 5 letters are in the correct envelopes, the 6th letter must also be in its correct envelope.

So,

x = 720 - 1 - 0 = 719

Step 2: Find $y$ (All letters in wrong envelopes)

This is a standard Derangement of 6 objects, denoted as $D_6$ . The formula for derangement of $n$ objects is:

D_n = n! \left[ \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \dots + \frac{(-1)^n}{n!} \right]

For $n = 6$ :

D_6 = 6! \left[ \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} + \frac{1}{6!} \right]

D_6 = 720 \left[ \frac{1}{2} - \frac{1}{6} + \frac{1}{24} - \frac{1}{120} + \frac{1}{720} \right]

D_6 = 360 - 120 + 30 - 6 + 1 = 265

So,

y = 265

Step 3: Find $x - y$

Now, we simply subtract the two values:

x - y = 719 - 265

x - y = 454

Final Answer:

The value of $x - y$ is 454.