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26

MULTIPLICATION OF SUMS

A proof of the binomial theorem

THE BINOMIAL THEOREM gives the coefficients in the product of n equal binomials:

(x + a)n = (x + a)(x + a)· · · (x + a).

If we actually multiplied the 4 factors of

(x + a)4,

then, before adding the like terms, we would find terms in

x4, x3a, x2a2, xa3, and a4.

The binomial theorem tells how many terms there are of each kind. Those binomial coefficients, the theorem states, are the combinatorial numbers.  To prove that, we will first consider the multiplication of any sums; for example,

(x + y)(a + b + c).

Upon multiplying, we would find six terms.  Each term will contain two factors, namely one letter from each factor:

xa + xb + xc + ya + yb + yc.

Therefore, we can write the product of the following --

(x + y)(a + b)(m + n)

-- simply by writing the sum of all combinations of one letter from each factor.

xam + xan + xbm + xbn + yam + yan + ybm + ybn.

Each term in the product consists of three factors:  one from each binomial.

Note that there are a total of 23 or 8 terms.  In general:

Multiplication of n binomials produces 2n terms.

For, multiplication of two binomials gives 4 terms:

(p + q)(m + n) =  pm + pn + qm + qn.

If we multiply those with a binomial, we will have 8 terms; those multiplied with a binomial will produce 16 terms; and so on.

Example.   (x + a)(x + b)(x + c)(x + d)

 =  x4  + (a + b + c + d)x3 + (ab + ac + ad + bc + bd + cd)x2
 
  + (abc + abc + acd + bcd)x + abcd.

For, each of the 24 terms will consist of 4 factors: one from each binomial.

x4 is produced by taking x from each factor.  xxxx = x4.  There is only one such term.  The coefficient of x4 is 1.

Terms with x3 are formed by taking  x  from any three factors, in every possible way, and the letter from the remaining factor.

axxx + xbxx + xxcx + xxxd = (a + b + c + d)x3.

The coefficient of x3, therefore, is the sum of the combinations of  a, b, c, d  taken 1 at a time:  a + b + c + d.

How many such combinations are there?

4C1: The number of combinations of 4 things taken 1 at a time.

Next, terms with x2 will come from taking x from two factors, in every possible way, and the letters from the remaining two. There will be 4C2 such terms: The number of ways of choosing 2 things -- 2 letters-- from 4.

(ab + ac + ad + bc + bd + cd)x2.

A term in x will be produced by taking x from 1 factor and the letter from the remaining 3:

(abc + abd + acd + bcd)x.

There will be 4C3 or 4 ways of doing that; of choosing 3 letters from 4.

Finally, the constant term will be produced by taking the letter from each of the 4 factors. There is 4C4 -- 1 -- way of doing that. The constant term will be

abcd.

Again:

(x + a)(x + b)(x + c)(x + d)

 =  x4  + (a + b + c + c)x3 + (ab + ac + ad + bc + bd + cd)x2
 
  + (abc + abc + acd + bcd)x + abcd.

If we now make all the constants equal -- a = b = c = d -- then we have the 4th power of (x + a ):

(x + a)4 = 4C0x4 + 4C1ax3 + 4C2a2x2 + 4C3a3x + 4C4a4
 
  = x4 + 4ax3 + 6a2x2 + 4a3x + a4.

The binomial coefficients are the number of terms of each kind. In the epansion of (x + a)n with n = 4, they are 1  4  6  4  1.

The result is general.  The binomial theorem states that in the expansion of (x + a)n, the coefficients are the combinatorial numbers nCk , where k -- the exponent of a -- successively takes the values 0, 1, 2, . . . , n.

  Each term in the expansion will have this form:          nExclamation!       
(nk)Exclamation! kExclamation!
 ankbk .

Compare Example 4, Lesson 25.

Problem 1.   Imagine multiplying out  (x + y + z)(a + b + c).

a)  How many terms would there be?   32 = 9

b)  Each term would consist of how many factors?   Two

Problem 2.   Multiply out by taking the correct combinations of the integers.

a)  (x + 1)(x + 3)(x + 4)  = x3 + 8x2 + 19x + 12

b) (x + 1)(x + 2)(x + 3)(x − 1)  = x4 + 5x3 + 5x2 − 5x − 6

Problem 3.   In this multiplication  (x + 1)(x + 2)(x + 3)(x + 4)(x + 5)  what will be the coefficient of x3?   85

Problem 4.   (x + a)5 = (x + a)(x + a)(x + a)(x + a)(x + a).

a)  Upon multiplying out, and before collecting like terms, how many
a)  terms will be produced?   25 = 32

b)  How will a term  x3a2  be produced?

By taking a from any two factors, in every possible way, and x from the remaining three factors.

c)  How many times will that term be produced?  In other words, upon
c)  adding those like terms, what number will be the coefficient of  x3a2?

The number of ways of choosing 2 things -- letter a -- from 5:

5C2 = 10

Problem 5.   In each row of Pascal's triangle, the sum of the binomial coefficients is 2n.  Why?

2n is the number of terms upon multiplying n binomials. Each binomial coefficient is the number of terms of that kind. Therefore, the sum of all the terms will be 2n.

End of the lessson

Next Topic:  Mathematical induction


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