Extended pigeonhole principle.

Let "n" be the number of pigeon and "m" be the number of pigeonholes and

If (m < < n) i.e. number of pigeonholes are less than the number of pigeons then extended pigeonhole principle is applicable

According to extended pigeonhole principle one pigeonhole must contains at-least

[$\frac {n-1}{m} + 1$] pigeons.

Here Number of pigeons = n = ?

No. of pigeonholes = m = 12 (months)

$\therefore$ $[\frac{n-1}{m}] + 1 = 5$

$[\frac{n-12}{m}] + 1 = 5$

n - 1 = 48

n = 49 [ No. of pigeons]

$\therefore$ 49 friends should be their to guarantee that at-least five of them must have birthday in a same month of year