state prove binomial theorem
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Answer:
Step-by-step explanation:
The binomial theorem states that in the expansion of (x + a) n, the coefficients are the combinatorial numbers nC k , where k -- the exponent of a -- successively takes the values 0, 1, 2, . . . , n
Answer:
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 + abd + 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 + abd + 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 expansion 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.