Dear John: Your discussion of the binomial theorem gave me the missing clue I needed to figure out what the "quiz lad" actually did in his simplified system of determining the coefficient of each term in any algebraic expression resulting from raising any simple binomial to any power.1
I told you he developed a pyramid-like structure of numbers, each row corresponding to the coefficients applicable to that particular exponent of the binomial.
First, the explanation:
(a + b) to the first power--
coefficients are 1, 1.
For each power we have an expression of one more term than the exponent itself; i.e., (a + b) has three terms. Now the building of the pyramid becomes easy according to the following rule. After placing the single digit, 1, at beginning and end of the expression, each coefficient of the intermediate terms is found by merely adding each consecutive pair of coefficients from the line above.
Here they are:
(a + b)1 = 1, 1
(a + b)2 = 1, 2, 1
(a + b)3 = 1, 3, 3, 1
(a + b)4 = 1, 4, 6, 4, 1
(a + b)5 = 1, 5, 10, 10, 5, 1
(a + b)6 = 1, 6, 15, 20, 15, 6, 1
(a + b)7 = 1, 7, 21, 35, 35, 21, 7, 1
etc.
Example:
Take (a + b)6--place 1 at beginning and end. There are five coefficients left to find. So examine the numbers in the fifth line. The first two add to 6. The second and third, add to 15. The third and fourth, add to 20--etc.
Simple!2 As ever
