To simplify the description
below, we will first define a "suit bid". A "suit bid" consists of a
number (1, 2, 3, 4, 5, 6, or 7) followed by a suit (Clubs, Diamonds,
Hearts, Spades, No Trump). We note that any of the 7 numbers may be
combined with any of the 5 suits yielding a choice of 35 possible bids.
The bidding process may use none, one, etc., up to 35 of these suit
bids. The words "Pass", "double", and "redouble" may also be used as
part of the bidding process.
The bidding process is divided into 3 groups:
1) Combinations before the first suit
bid.
2) Combinations using suit bids.
3) Combinations after suit bids have concluded.
For each quantity of suit bids (1-35), the totals for all
3 groups are multiplied together. The grand total is the sum of these
plus 1 (For pass, pass, pass, pass).
There are only 4 combinations for group "1)" which are:
Bid
Pass, Bid
Pass, Pass, Bid
Pass, Pass, Pass, Bid
Group "3)" is nearly as simple with only 7 combinations.
Bid, Pass, Pass, Pass
Bid, Double, Pass, Pass, Pass
Bid, Pass, Pass, Double, Pass, Pass, Pass
Bid, Double, Redouble, Pass, Pass, Pass
Bid, Double, Pass, Pass, Redouble, Pass, Pass, Pass
Bid, Pass, Pass, Double, Redouble, Pass, Pass, Pass
Bid, Pass, Pass, Double, Pass, Pass, Redouble, Pass, Pass, Pass
Group "2)" is significantly more complicated. There are 35
possible suit bids (Digits 1 - 7 times 5 suits). Thus, if we only use
one of these "suit bids", there are COMBIN( 35, 1) = 35 possible
combinations. If we use any 2 "suit bids", then there are COMBIN( 35,
2) = 595 combinations. (Take any 2 from 35). 3 "suit bids" yields
COMBIN( 35, 3) = 6,545. This process repeats up through COMBIN( 35, 35)
= 1.
In-between each of the suit bids, there are 21 possible intervening
sequences:
Bid, Bid
(No intervening "Passes", "doubles", "redoubles")
Bid, Pass, Bid
Bid, Pass, Pass, Bid
Bid, Double, Bid
Bid, Double, Pass, Bid
Bid, Double, Pass, Pass, Bid
Bid, Pass, Pass, Double, Bid
Bid, Pass, Pass, Double, Pass, Bid
Bid, Pass, Pass, Double, Pass, Pass, Bid
Bid, Double, Redouble, Bid
Bid, Double, Redouble, Pass, Bid
Bid, Double, Redouble, Pass, Pass, Bid
Bid, Double, Pass, Pass, Redouble, Bid
Bid, Double, Pass, Pass, Redouble, Pass, Bid
Bid, Double, Pass, Pass, Redouble, Pass, Pass, Bid
Bid, Pass, Pass, Double, Redouble, Bid
Bid, Pass, Pass, Double, Redouble, Pass, Bid
Bid, Pass, Pass, Double, Redouble, Pass, Pass, Bid
Bid, Pass, Pass, Double, Pass, Pass, Redouble, Bid
Bid, Pass, Pass, Double, Pass, Pass, Redouble, Pass, Bid
Bid, Pass, Pass, Double, Pass, Pass, Redouble, Pass, Pass, Bid
If there is only 1 "suit bid", the above intervening sequence is
used 0 times.
If there are 2 "suit bids", the above intervening sequence is
used 1 time.
If there are 3 "suit bids", the above intervening sequence is
used 2 times.
etc.
If there are 35 "suit bids", the above intervening sequence is
used 34 times.
Thus the mathematical calculation for the number of combinations for
group 2) becomes:
Number of Mathematical
Suit Bids Expression
------------------------------
1
COMBIN( 35, 1) * 21^0 = 35 * 1 = 35
2
COMBIN( 35, 2) * 21^1 = 595 * 21 = 12,495
3
COMBIN( 35, 3) * 21^2 = 6,545 * 441 = 2,886,345
etc.
35 COMBIN( 35,
35) * 21^34 = 9.025 E+44
Each of the above numbers is then multiplied by 4 for group 1) and then
multiplied again by 7 for group 3). The result of all this generates
the numbers that appear in the Stats table.
At this point we call in the computer. A simplified "C" program might
look like:
Coef =
28.0;
/* Init coef with Group 1) times Group 3) */
TotComb =
1.0;
/* Init count with "Pass, Pass, Pass, Pass" */
/* For 1 through 35 "suit bids"
*/
for (i = 1, j = 35; i <= 35;
i++, j--) {
Coef *=
j;
/* Update the coef for the COMBIN()
*/
Coef /=
i;
/*
function.
*/
printf( "
%2d %g\n", i, Coef); /* Output the next
row in the result */
TotComb +=
Coef;
/* Update the grand
total
*/
Coef *=
21.0;
/* 21 new intervening
comb.
*/
}
Alternately a spreadsheet could be used (Will only require 35 rows).
And that's it. The output isn't formatted perfectly, but we'll leave
that to the reader.
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