Given below is the percentage distribution of the number of students from 5 different colleges who attended different numbers of seminars.

| College | Attended 1 Seminar (%) | Attended 2 Seminars (%) | Attended 3 Seminars (%) | Attended 4 Seminars (%) | Attended 5 Seminars (%) |
|---------|-----------------------|------------------------|------------------------|------------------------|------------------------|
| P | 10 | 12 | 15 | 17 | 13 |
| Q | -- | 36 | 52 | -- | 11 |
| R | 22 | 38 | 25 | -- | 15 |
| S | -- | 28 | 32 | 17 | 10 |
| T | 35 | 70 | 72 | 32 | 5 |

If the number of students from college R who attended at most 3 seminars is equal to the number of students from college S who attended at least 3 seminars, and the total number of students from college S is 11400, then find the number of students from college R who attended exactly 3 seminars.

Question

Given below is the percentage distribution of the number of students from 5 different colleges who attended different numbers of seminars.

| College | Attended 1 Seminar (%) | Attended 2 Seminars (%) | Attended 3 Seminars (%) | Attended 4 Seminars (%) | Attended 5 Seminars (%) |
|---------|-----------------------|------------------------|------------------------|------------------------|------------------------|
| P       | 10                    | 12                     | 15                     | 17                     | 13                     |
| Q       | --                    | 36                     | 52                     | --                     | 11                     |
| R       | 22                    | 38                     | 25                     | --                     | 15                     |
| S       | --                    | 28                     | 32                     | 17                     | 10                     |
| T       | 35                    | 70                     | 72                     | 32                     | 5                      |

If the number of students from college R who attended at most 3 seminars is equal to the number of students from college S who attended at least 3 seminars, and the total number of students from college S is 11400, then find the number of students from college R who attended exactly 3 seminars.

Accepted Answer

**Solution:**

1. First, calculate the number of students from college S who attended at least 3 seminars.
   - At least 3 seminars means students who attended 3, 4, or 5 seminars.
   - Percentage for college S: 32% (3 seminars) + 17% (4 seminars) + 10% (5 seminars) = 59%.

2. Total students in college S = 11400.
   - Number of students attending at least 3 seminars from S = 59% of 11400 = $0.59 \times 11400 = 6726$.

3. According to the problem, the number of students from college R who attended at most 3 seminars equals this number.
   - At most 3 seminars means students who attended 1, 2, or 3 seminars.
   - Percentage for college R: 22% (1 seminar) + 38% (2 seminars) + 25% (3 seminars) = 85%.

4. Let the total number of students in college R be $x$.
   - Then, 85% of $x$ = 6726
   - $0.85x = 6726 \Rightarrow x = \frac{6726}{0.85} = 7912.94 \approx 7913$.

5. Number of students from college R who attended exactly 3 seminars = 25% of total students in R = $0.25 \times 7913 = 1978.25$.

6. However, the options given are much larger, so let's re-check the problem statement carefully.

**Re-examining the problem:**
- The problem states: "number of students from college R who attended at most 3 seminars is equal to number of students from college S who attended at least 3 seminars."
- We calculated the number of students from S who attended at least 3 seminars as 6726.
- So, number of students from R who attended at most 3 seminars = 6726.
- Percentage of students from R who attended at most 3 seminars = 22% + 38% + 25% = 85%.

Therefore, total students in R = $\frac{6726}{0.85} = 7913$.

Number of students from R who attended exactly 3 seminars = 25% of 7913 = 0.25 \times 7913 = 1978.25.

Since this value does not match any option, let's consider if the percentages are absolute numbers or if the problem expects a different approach.

**Alternate approach:**
- The table shows percentages, but the problem might be assuming the percentages are out of different totals.
- The problem states "percentage distribution," so the percentages are relative to each college's total students.

Given that, the calculation above is correct.

**Conclusion:**
- Number of students from college R who attended exactly 3 seminars = approximately 1978.
- Since this is not among the options, check if the problem expects the number of students from R who attended exactly 3 seminars to be equal to the number of students from S who attended at least 3 seminars.

If the problem states "number of students from college R who attended at most 3 seminars is equal to number of students from college S who attended at least 3 seminars," then the number of students from R who attended exactly 3 seminars is:

$$
\text{Students in R attending exactly 3 seminars} = 25\% \times \text{Total students in R} = 25\% \times \frac{6726}{85\%} = 25\% \times 7913 = 1978
$$

Since 1978 is not an option, possibly the problem expects the number of students from R who attended exactly 3 seminars to be equal to the number of students from S who attended exactly 3 seminars.

Number of students from S who attended exactly 3 seminars = 32% of 11400 = 0.32 \times 11400 = 3648.

This also does not match options.

**Reconsidering the problem statement:**
- The problem says: "number of students from college R who attended at most 3 seminars is equal to number of students from college S who attended at least 3 seminars and the total number of students from College S is 11400."
- We found total students in R = 7913.
- Number of students from R who attended exactly 3 seminars = 25% of 7913 = 1978 (approx).

Since none of the options match, the closest option to 1978 is none of these.

**But the user input options are:**
(a) 3700 (b) 4100 (c) 4500 (d) 4800 (e) None of these

**Hence, the answer is (e) None of these.**

**Answer:** None of these

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