Four different objects 1, 2, 3, 4 are distributed at random in four places marked 1, 2, 3, 4. What is the probability that none of the objects occupy the place corresponding to its number ?
- \(\frac{17}{24}\)
- \(\frac{3}{8}\)
- \(\frac{1}{2}\)
- \(\frac{5}{8}\)
The Correct Option is B
Solution and Explanation
The Number of derangement's of set with n elements is
\(Dn = n![1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - .............+ (-1)^n \frac{1}{n!}]\)
Probability of derangement = \(\frac{Dn}{n!}\)
From the case given above the value of \(n\) = 4
So, The Probability that none of the objects occupy the place corresponding to their number
= \(1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!}\)
= \(1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24}\)
= \(\frac{12 - 4 + 1}{24}\) = \(\frac{9}{24}\)
= \(\frac{3}{8}\)
The correct option is (B): \(\frac{3}{8}\)
Top Questions on Probability
Based upon the results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very healthy, 200 maintained average health and 100 had a poor health record.
Let \( A_1 \): People with good health,
\( A_2 \): People with average health,
and \( A_3 \): People with poor health.
During a pandemic, the data expressed that the chances of people contracting the disease from category \( A_1, A_2 \) and \( A_3 \) are 25%, 35% and 50%, respectively.
Based upon the above information, answer the following questions:
(i) A person was tested randomly. What is the probability that he/she has contracted the disease?}
(ii) Given that the person has not contracted the disease, what is the probability that the person is from category \( A_2 \)?- If A and B are two events such that $P(B) = \frac{1}{5}$, $P(A | B) = \frac{2}{3}$ and $P(A \cup B) = \frac{3}{5}$, then $P(A)$ is :
- If \( P(A) = \frac{1}{5}, P(B) = \frac{3}{5} \) and \( P(A \cap B) = \frac{2}{5} \), then \( P(A' \cup B') \) is :
- Bag I contains 4 white and 5 black balls. Bag II contains 6 white and 7 black balls. A ball drawn randomly from Bag I is transferred to Bag II and then a ball is drawn randomly from Bag II. Find the probability that the ball drawn is white.
- A meeting will be held only if all three members A, B and C are present. The probability that member A does not turn up is 0.10, member B does not turn up is 0.20 and member C does not turn up is 0.05. The probability of the meeting being cancelled is:
Questions Asked in MAT exam
- An international NGO for homeless people has supplied certain items for ongoing winters to new shelters for homeless people. These consist of different items like blankets, jackets, shoes, socks, and room heaters. The total number of five items distributed in December 2024 was 3300.
- 24% of all items were blankets.
- $\frac{1}{6}$ of all items were jackets.
- 14% of all items were shoes.
- Remaining items were either socks or room heaters.
- The number of room heaters distributed was 100 more than the number of socks distributed.
Find the ratio between the number of room heaters and the number of shoes?- MAT - 2025
- Data Analysis
- An international NGO for homeless people has supplied certain items for ongoing winters to new shelters for homeless people. These consist of different items like blankets, jackets, shoes, socks, and room heaters. The total number of five items distributed in December 2024 was 3300.
- 24% of all items were blankets.
- $\frac{1}{6}$ of all items were jackets.
- 14% of all items were shoes.
- Remaining items were either socks or room heaters.
- The number of room heaters distributed was 100 more than the number of socks distributed.
The number of socks distributed is approximately what percent of the total number of shoes and room heaters put together?- MAT - 2025
- Data Analysis
- An international NGO for homeless people has supplied certain items for ongoing winters to new shelters for homeless people. These consist of different items like blankets, jackets, shoes, socks, and room heaters. The total number of five items distributed in December 2024 was 3300.
- 24% of all items were blankets.
- $\frac{1}{6}$ of all items were jackets.
- 14% of all items were shoes.
- Remaining items were either socks or room heaters.
- The number of room heaters distributed was 100 more than the number of socks distributed.
What is the total number of jackets, shoes, and room heaters distributed?- MAT - 2025
- Data Analysis
- The Chatterjee family has 8 members: A, B, C, D, E, F, G, H. There are 3 generations in the family. They go to 4 different countries: France, Canada, UK, and USA. They make 4 groups of 2 members each, and each group goes to one country. The family consists of 3 married couples, and each couple forms a group. The following conditions are given:
1. The family consists of 3 married couples and each couple forms a group.
2. The oldest member of the family goes to Canada.
3. H, a female, is married to E, who does not go to UK.
4. B and Dform a group and agreed to go to USA.
5. E’s mother-in-law is the mother of C, who is the father of B and brother of H.
6. A is the grandfather of one of the male members, who goes to USA.
7. D is unmarried and the niece of C.
8. F is the oldest member of the family.
Which of the following members will go to Canada?- MAT - 2025
- Family Tree Problems
- An international NGO for homeless people has supplied certain items for ongoing winters to new shelters for homeless people. These consist of different items like blankets, jackets, shoes, socks, and room heaters. The total number of five items distributed in December 2024 was 3300.
- 24% of all items were blankets.
- $\frac{1}{6}$ of all items were jackets.
- 14% of all items were shoes.
- Remaining items were either socks or room heaters.
- The number of room heaters distributed was 100 more than the number of socks distributed.
What is the difference between the total number of socks and blankets distributed and the number of shoes distributed?- MAT - 2025
- Data Analysis