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A091673 Numerator Q of probability P=Q(n)/365^(n-1) that exactly two out of n people share the same birthday. 2
1, 1092, 793884, 480299820, 261163522620, 132358677731280, 63798093049771080, 29612552769907347240, 13345042642324219106280, 5872442544965392834838400, 2533775368098060137659608000, 1075256447734638237381213700800 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

A 365-day year and a uniform distribution of birthdays throughout the year is assumed.

LINKS

Table of n, a(n) for n=2..13.

P. Le Conte, Coincident Birthdays.

Eric Weisstein's World of Mathematics, Birthday Problem. Section in World of Mathematics.

FORMULA

P(n) = n!*Sum_{i=1..floor(n/2)}(binomial(365, i)*binomial(365-i, n-2*i)/2^i).

EXAMPLE

a(3)=1092 because the probability in a group of 3 people that exactly two of them share the same birthday is (1/365^3)*3!*binomial(365,1)*binomial(364,1)/2=

(1/365^2)*3*364=(1/365^2)*1092.

MATHEMATICA

P[n_] := (n! Sum[ Binomial[365, i]*Binomial[365 - i, n - 2i] /2^i, {i, 1, Floor[n/2]}]/365); Table[ P[n], {n, 2, 13}] (* Robert G. Wilson v, Feb 09 2004 *)

CROSSREFS

Cf. A014088, A091674 gives probabilities for two or more coincidences, A091715 gives probabilities for three or more coincidences.

Sequence in context: A043873 A240719 A239875 * A288097 A281001 A271100

Adjacent sequences:  A091670 A091671 A091672 * A091674 A091675 A091676

KEYWORD

frac,nonn

AUTHOR

Hugo Pfoertner, Feb 03 2004

EXTENSIONS

More terms from Robert G. Wilson v, Feb 09 2004

Broken links corrected by Steven Finch, Jan 27 2009

STATUS

approved

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Last modified July 26 20:17 EDT 2021. Contains 346294 sequences. (Running on oeis4.)