To solve the problem, let's define the sets for clarity:
- Let \( E \) be the set of people who read English newspapers.
- Let \( H \) be the set of people who read Hindi newspapers.
- Let \( R \) be the set of people who read Regional Language newspapers.
From the problem, we have the following values:
- \( |E| = 21 \) (read English newspapers)
- \( |H| = 26 \) (read Hindi newspapers)
- \( |R| = 29 \) (read Regional Language newspapers)
- \( |E \cap H| = 14 \) (read both English and Hindi newspapers)
- \( |H \cap R| = 15 \) (read both Hindi and Regional Language newspapers)
- \( |E \cap R| = 12 \) (read both English and Regional Language newspapers)
- \( |E \cap H \cap R| = 8 \) (read all three newspapers)
To find the number of people surveyed and those who read only the regional language paper, we'll apply the principle of inclusion-exclusion and then subtract the overlaps properly.
Part (a): Find the total number of people surveyed
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Using the Inclusion-Exclusion Principle:
The total number of people who read at least one type of newspaper is given by:
\[ |E \cup H \cup R| = |E| + |H| + |R| - |E \cap H| - |H \cap R| - |E \cap R| + |E \cap H \cap R| \]
Substituting the values:
\[ |E \cup H \cup R| = 21 + 26 + 29 - 14 - 15 - 12 + 8 \]
Calculating step-by-step:
\[ = 21 + 26 + 29 = 76 \] \[ = 76 - 14 - 15 - 12 + 8 \] \[ = 76 - 41 + 8 = 43 \]
So, the total number of people surveyed is \( \boxed{43} \).
Part (b): Find how many people read only the regional language paper
To find how many people read only the regional language newspaper \( |R_{\text{only}}| \):
\[ |R_{\text{only}}| = |R| - (|H \cap R| + |E \cap R| - |E \cap H \cap R|) \]
Substituting known values:
\[ |H \cap R| = 15, \quad |E \cap R| = 12, \quad |E \cap H \cap R| = 8 \]
Calculating
\[ |R_{\text{only}}| = 29 - (15 + 12 - 8) \]
Calculating the overlaps:
\[ = 29 - (15 + 12 - 8) = 29 - 19 = 10 \]
So, the number of people who read only the regional language newspaper is \( \boxed{10} \).
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