## Teaser 2941: Big deal

**From The Sunday Times, 3rd February 2019** [link]

My wife and I play bridge with Richard and Linda. The 52 cards are shared equally among the four of us – the order of cards within each player’s hand is irrelevant but it does matter which player gets which cards. Recently, seated in our fixed order around the table, we were discussing the number of different combinations of cards possible and we calculated that it is more than the number of seconds since the “big bang”!

We also play another similar game with them, using a special pack with fewer cards than in the standard pack – again with each player getting a quarter of the cards. Linda calculated the number of possible combinations of the cards for this game and she noticed that it was equal to the number of seconds in a whole number of days.

How many cards are there in our special pack?

[teaser2941]

## Jim Randell 12:32 pm

on1 March 2019 Permalink |If there are

kplayers, and each player receivesncards in the deal, then there aren.kcards in total.The number of ways of dealing

ncards to the first player from the total set ofn.kcards are:For the second player there are only

n(k – 1)cards remaining, and dealingncards from that we get:For the third player:

and so on up to the final player

Multiplying all these together we find that one of the terms in the denominator cancels with the numerator for the next player, giving a total number of possible deals of:

We can use this rather neat formula to find the required number of cards in the puzzle.

This Python program runs in 74ms.

Run:[ @repl.it ]Solution:There are 28 cards in the special pack.LikeLike