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CSAT 2023

Q6 2023, Probability

Consider the following statements regarding the random insertion of four letters into four envelopes, where each letter is to be placed in the envelope with the correct address:
1. It is possible that exactly one letter goes into an incorrect envelope.
2. There are only six ways in which only two letters can go into the correct envelopes.
Which of the statements given above is/are correct?

चार पत्र और चार लिफाफे हैं और ठीक-ठीक एक पत्र को सही पते वाले ठीक-ठीक एक लिफाफे में डालना है । यदि पत्रों को लिफाफों में यादृच्छिक रूप से डाला जाता है, तो निम्नलिखित कथनों पर विचार कीजिए:
1. यह संभव है कि ठीक-ठीक एक पत्र गलत लिफाफे में जाए ।
2. ऐसे केवल छह तरीके हैं जिनमें केवल दो पत्र ही सही लिफाफे में जा सकते हैं ।
उपर्युक्त कथनों में से कौन-सा/से सही है/हैं ?

Explanation Please wait..

Statement 1 is incorrect.

If a letter is put in the wrong envelope, the letter that should have gone in that envelope must also be in the wrong envelope.

Therefore, it is not possible for only one letter to be misplaced.

Either no letter will be misplaced, or at least two letters will be misplaced.

Calculations for Statement 2:

Number of ways in which only two letters can go into the correct envelopes:

To have exactly two letters in the correct envelopes, we need to consider the arrangements where two letters are placed correctly and the other two are not.

The number of ways to choose two letters out of four is given by the combination formula

where 𝑛 is the total number of items (letters in this case) and 𝑘is the number of items to be chosen (2 letters here).

There are 6 ways to choose two letters from four.

However, in each of these selections, there is only one correct arrangement (as the other two letters are fixed in their incorrect envelopes).

Therefore, there are only 6 ways in which exactly two letters can go into the correct envelopes, as stated in statement 2.

Hence, statement 2 is correct.

Statement 1 is incorrect.

<ul><li> 
If
a letter is put in the wrong envelope, the letter that should have gone in that
envelope must also be in the wrong envelope.</li></ul>

<ul><li> 
Therefore,
it is not possible for only one letter to be misplaced.</li></ul>

<ul><li> 
Either
no letter will be misplaced, or at least two letters will be misplaced.</li></ul>

Calculations for Statement 2:

Number of ways
in which only two letters can go into the correct envelopes:

<ul><li> 
To
have exactly two letters in the correct envelopes, we need to consider the
arrangements where two letters are placed correctly and the other two are not.</li></ul>

<ul><li> 
The
number of ways to choose two letters out of four is given by the combination
formula</li></ul>

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<ul><li> 
where
𝑛 is the
total number of items (letters in this case) and 𝑘is
the number of items to be chosen (2 letters here).</li></ul>

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<ul><li> 
There
are 6 ways to choose two letters from four.</li></ul>

<ul><li> 
However,
in each of these selections, there is only one correct arrangement (as the
other two letters are fixed in their incorrect envelopes).</li></ul>

<ul><li> 
Therefore,
there are only 6 ways in which exactly two letters can go into the correct
envelopes, as stated in statement 2.</li></ul>

<ul><li> 
Hence,
statement 2 is correct.</li></ul>

CSAT Year Wise (Temporary Link, This will be removed and no PDFs shall be downloaded here. Please download CSAT subject wise files) / सी सैट

Q6 2023, Probability

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CSAT Year Wise (Temporary Link, This will be removed and no PDFs shall be downloaded here. Please download CSAT subject wise files) / सी सैट

Q6 2023, Probability

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