Henry Ernest Dudeney/Modern Puzzles/91 - Counting the Wounded/Solution

by : $91$

 * Counting the Wounded

Solution
$120$ men are in the hospital.

Proof
We know that:
 * exactly two-thirds of the men had lost an eye.

We also know that:
 * three-fourths had lost an arm.

By Inclusion-Exclusion Principle, at least:
 * $\dfrac 2 3 + \dfrac 3 4 - 1 = \dfrac 5 {12}$

of the men had lost both an eye and an arm.

Similarly, since we know that:
 * four-fifths had lost a leg

by the Inclusion-Exclusion Principle again, at least:
 * $\dfrac 5 {12} + \dfrac 4 5 - 1 = \dfrac {13} {60}$

of the men had lost all three.

equates this to $26$.

Hence there are exactly:
 * $26 \div \dfrac {13} {60} = 120$

men in this hospital.