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The rules for the University of Bombay five-a-side cricket competition specify that the members of each team must have birthdays in the same month. What is the minimum number of mathematics students needed to be enrolled in the department to guarantee that they can raise a team of students?

  1. $23$
  2. $91$
  3. $60$
  4. $49$
  5. None of the above
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3 Answers

Best answer
28 votes
28 votes

There are ${12}$ months and we have to get $5$ people having birthdays in the same month in order to form a team. Pigeon hole principle can be applied here :

$\lceil \frac{N}{12} \rceil= 5$,

$N=49 .$

Hence answer is D

6 votes
6 votes
Jan Feb March April May June July August September October November December
${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$   ${\color{DarkRed} S}$   ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$
${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$   ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$
${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$   ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$
${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$   ${\color{Magenta} S}$ ${\color{Magenta} S}$   ${\color{Magenta} S}$

Procedure:-

Select $12$ students and then put them in slots of $1^{st}$ row each like $1^{st}$ in jan, $2^{nd}$ in feb ,$3^{rd}$ in march and so on.

Repeat the above step $4$ times.

so  Total students selected till now $= 12*4 =48$ and in each month we have $4$ students.

now if we select $1$ more student then his birthday can be in any of the given $12$ months.

say his birthday is in December then we can make a team of $5$ students such that

the members of the team must have birthdays in the same month.

$\therefore$ Total students required $= 48+1 =49.$

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