Read the information given below carefully and answer the question that follows:

(A) If the mean of 5 numbers is 15 and then if 27 is included to the numbers, then the mean of the 6 numbers shall be 17
(B) The mean of marks of 300 students is 60, the mean of the top 100 students' marks is 80 and the mean of the last 100 students is 50, then the mean of the marks of the rest of the 100 students of the group is 60.
(C) The mean and median of 5 observations is 9 and 8 respectively, if 1 is subtracted from each observation, then the new mean and median shall be 8 and 7 respectively

Choose the correct answer from the options given below:

(A) and (C) Only

The correct answer is Option (4) → (A) and (C) Only

Statement (A) Verification

• Initial data: Mean of 5 numbers is 15.
• Sum of 5 numbers $= 5 \times 15 = 75$.
• New data: 27 is included.
• New sum $= 75 + 27 = 102$.
• New count $= 6$.
• New Mean $= \frac{102}{6} = 17$.
• Conclusion: Statement (A) is Correct.

Statement (B) Verification

• Total Sum: 300 students with a mean of 60.
• Total sum $= 300 \times 60 = 18,000$.
• Sum of Groups:
• Top 100 students (mean 80): $100 \times 80 = 8,000$.
• Last 100 students (mean 50): $100 \times 50 = 5,000$.
• Remaining 100 students: $18,000 - (8,000 + 5,000) = 5,000$.
• Mean of rest: $\frac{5,000}{100} = 50$.
• The statement claims the mean of the rest is 60.
• Conclusion: Statement (B) is Incorrect.

Statement (C) Verification

• Property of Mean: If a constant $k$ is subtracted from every observation, the new mean is the original mean minus $k$.
• Original Mean $= 9$. New Mean $= 9 - 1 = 8$.
• Property of Median: If a constant $k$ is subtracted from every observation (preserving their relative order), the new median is the original median minus $k$.
• Original Median $= 8$. New Median $= 8 - 1 = 7$.
• Conclusion: Statement (C) is Correct.

Final Result:

Statements (A) and (C) are correct.