Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set :

Question

If \( A = \{1,2,3,4,5,6\} \) and \( B = \{1,2,3,\ldots,9\} \), then the number of strictly increasing functions \( f : A \to B \) such that \( f(i) \neq i \) for all \( i = 1,2,3,4,5,6 \) is:

Answer 1

To solve this problem, we need to determine the range of possible values for the percentage \( K \) of patients suffering from both heart ailments and lung infections. We are given:

89% of the patients are suffering from heart ailments.
98% of the patients are suffering from lung infections.
We need to find the value of \( K \) such that the set of possible values does not include \( K \).

Let's denote:

\( H = 89\%\) representing the percentage of patients with heart ailments.
\( L = 98\%\) representing the percentage of patients with lung infections.
\( K \) representing the percentage of patients with both ailments.

The principle of inclusion-exclusion for set theory gives us:

\[ |H \cup L| = |H| + |L| - |H \cap L| \]

In terms of percentages:

\[ \text{Total Percentage with at least one ailment} = H + L - K \]

Since the total percentage cannot exceed 100%:

\[ 89 + 98 - K \leq 100 \]

Simplifying gives:

\[ K \geq 187 - 100 = 87 \]

This implies that \( K \) must be at least 87%. Now let's rule out the sets:

\{84, 86, 88, 90\} — \(87\) and higher is not ruled out, as 88 and 90 are possible.
\{80, 83, 86, 89\} — \(87\) and higher is not ruled out, as 89 is possible.
\{79, 81, 83, 85\} — All values are below 87, so this set cannot include \( K \).
\{84, 87, 90, 93\} — \(87\) and higher is not ruled out, as 87, 90, and 93 are possible.

Therefore, the correct answer is that the percentage of patients suffering from both ailments, \( K \), cannot belong to the set \{79, 81, 83, 85\}.

Answer 2