Each line in Pascal's Triangle is derived from the line above. Each integer in a lower line equals the sum of two integers (one to the left and one to the right) in the immediate higher line. The triangle can continue down for as many lines as you want. The triangle can be traced back to Chinese mathematicians of 1303 who published the diagram as The Precious Mirror of the Four Elements (what a glorious name).
It is rather fitting that such a beautiful work of mathematics is relevant to the equally beautiful game of bridge. Try adding up the integers in each row. They add up to 1, 2, 4, 8, 16, 32, 64, 128 which you will recognise as 20, 21, 22, 23, 24, 25, 26, 27 etc.You may recognise 2 * 2 as the number of ways that two cards can be divided between two hands. The first card can be dealt to one of two hands, and the second card can be dealt to one of two hands. Hence 2 * 2. Similarly 2 * 2 * 2 is the number of ways that three cards can be divided between two hands. Each of the three hands can be placed in one of two hands.
*** 2n is the number of ways in which n cards can be distribued between two hands.
Let us look more closely at row n. Column k (the first column is column 0) is the way of chosing k cards from n cards. (The mathematics - n! / k! (n-k)! where n! is n factorial = n * (n-1) * (n-2 ) * ..* 2 * 1).
Looking at row 6
k=0 6!(0!6!) = 1/0! = 1/1 =1
k=1 6!(1!5!) = 6
k=2 6!(2!4!) = 6 * 5 /2 =15
k=3 6!(3!3!) = (6 * 5 * 4)/(3*2*1) = 20
k=4 6!/(4!2!) same as k=2 (15)
k=5 6!/(5!1!) same as k=1 (6)
k=6 6!(6!0!) same as k=0 (1)
Hopefully now the connection of the triangle to bridge is becoming apparent.
You are South and declarer. You and dummy have 7 spades, East and West therefore have 6 spades between them, and we now know there are 26 = 64 combinations of how these cards lie between East and West.
Now read across row 6
East has 0 cards - Column 0 1 combination (only 1 way of holding 0)
East has 1 card - Column 1 6 combination (can hold 6 possible cards)
East has 2 cards - Column 2 15 combinations ((6 * 5) /2 - (1st card from 6 * 2nd card from 5) / 2 (as choosing card A followed by card B gives the same combination as Card B followed by card A))
East has 3 cards - Column 3 20 combinations (6 * 5 * 4 /6 - (1st card from 6 * 2nd card from 5 * 3rd card from 4) / 6 (combination ABC can be chosen as ABC, ACB, BAC, BCA, CAB, CBA))
East has 4 cards - Column 4 15 combinations
East has 5 cards - Column 5 6 combinations
East has 6 cards - Column 6 1 combination
Total 64 combinations
The probability of a 3-3 break is therefore 20 / 64 = 31%
The probability of a 4-2 break (either way) is 2 * 15 / 64 = 47%
The probability of a 5-1 break (either way) is 2 * 6/ 64 = 19%
The probability of a 6-0 break (either way) is 2 * 1/64 = 3%
*** if n is the number of cards that the opponents hold then in row n column m / 2n is the probability of an opponent holding m of those cards.
Does this have any practical use?
If you have difficulty remembering those missing card distribution probabilities then if you can remember Pascal's Triangle you have a means of calculating those probabilities at the table. What the triangle does is emphasize that when the number of missing cards an even number then the chance of an even split is less than the chance of a slightly more distributional split. (e.g when n =4 the combinations of a 2-2 split are 6 but the combinations of a 1-3 or 3-1 split total 8, so a 3-1 split (either way) is more probable)
There are some who may have spotted the probabilty of two cards splitting 1-1 is actually quoted as 52% 2-0 is 48%. Pascal's Triangle gives 50%. What we are saying here is that Pascal's Traingle gives you a quick way of getting a pretty good guide to the correct probability. The reason why the quoted probabilities are different will be the subject of a separate article.