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The Birthday Problem

If the number of persons in the room is 1:

The person has 365 chances out of 365 that his birthday is unduplicated:

    365
    365

If the number of persons in the room is 2:

The second person has 364 chances out of 365 of having a birthday different from Person 1.

    365  X   364
    365       365

Person number 3 has 363 chances out of 365 of having a birthday different from either of the first two.

365  X   364  X   363
365       365        365

We are multiplying the separate probabilities of these independent events because we want to obtain the probability that there are no duplicated birthdays.

Carry out the multiplication on the top line (numerator) as many times as there are people in the room. 365! / (365-N)!

Carry out the multiplication on the bottom line (denominator) as many times as there are people in the room. 365N

Divide and subtract from 1 to obtain the probability of at least one duplicated birthday.

Here's how to set it up in a spreadsheet:

First row gets five labels:

A1    N
B1    Days
C1    Days!
D1    Days^N
E1    P of match

Second row gets three values and two formulae

A2    1
B2    365
C2    365
D2    =365^A2
E2    =1-C2/D2

Third row gets three formulae

A3    =A2+1
B3    =B2-1
C3    =C2*B3

Copy as needed