Binomial Theorem (Part-II)

by Anurag Chandra

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Greatest Term:
- In a binomial expansion greatest term means numerically greatest term.
  $\therefore$ greatest term in greatest term in
- If rth term is the greatest term, then $\frac {t_r}{t_{r+1}}\underline >1$ and $\frac {t_r}{t_{r-1}}\underline >1$
  But if there is only one greatest term, then for rth term to be greatest term $\frac {t_r}{t_{r+1}}\underline >1$ and $\frac {t_r}{t_{r-1}}\underline >1$
- In order to find the greatest term in the expansion of , calculate $r=\frac {(n+1)|x|}{|x|+1}$
- - If r is not an integer, and k be the integer just less than r, then th term is the greatest term.
  - If r is an integer, then rth and th terms are the greatest terms and they are equal.
- $(1+x)^n=^nC_0+^nC_1x+^nC_2x^2+\cdots +^nC_nx^n \qquad \qquad \cdots (1)$
  Putting x = 1, we get
  $2^n=^nC_0+^nC_1+^nC_2+^nC_3+\cdots +^nC_n \qquad \qquad \cdots(2)$
  Putting x = â€"1 in (1), we get
  $0=^nC_0-^nC_1+^nC_2-^nC_3+\cdots +(-1)^n.^nC_n \qquad \qquad \cdots (3)$
  $(2)+(3)\Rightarrow 2n=2[^nC_0+^nC_2+^nC_4+\cdots ]$
  $\Rightarrow 2^{n-1}=^nC_0+^nC_2+^nC_4+ \qquad \qquad \cdots (4)$
  Here we cannot write the last term on R.H.S. unless it is known whether n is odd or even.
  $(2)-(3)\Rightarrow 2n=2[^nC_1+^nC_3+^nC_5+\cdots]$
  $\Rightarrow 2^{n-1}=^nC_1+^nC_3+^nC_5+ \qquad \qquad \cdots (5)$
  Thus,
  (i) $^nC_0+^nC_1+^nC_2+\cdots +^nC_n=2n$
  (ii) $^nC_0- ^nC_1+^nC_2-\cdots +(-1)^{n-1}$
  (iii) $^nC_0+^nC_2+^nC_4+\cdots = 2^{n-1}$
  (iv) $^nC_1-^nC_3+^nC_5\cdots =2^{n-1}$
- Note:
  (i) $^{10}C_0+^{10}C_2+^{10}C_4+\cdots +^{10}C_{10}=2^{10-1}=2^9$
  (ii) $^{10}C_1+^{10}C_3+^{10}C_5+\cdots +^{10}C_9=2^{10-1}=2^9$
  (iii) $^{11}C_0+^{11}C_2+^{11}C_3+\cdots +^{11}C_{10}=2^{11-1}=2^{10}$
  (iv) $^{11}C_1+^{11}C_3+^{11}C_5+\cdots +^{11}C_{11}=2^{11-1}= 2^{10}$
  Here last term is nCn if in each term upper and lower suffices are both even or both odd.
  Last term is nCnâ€"1 if in each term one of lower and upper suffices is even and other is odd.
- If Cr stands for , then
  - $C_0C_r+C_1C_{r+1}+C_2C_{r+2}+\cdots +C_{n-r}C_n=^{2n}C_{n-r}$
  - $C_0^2+C_1^2+C_2^2+\cdots C_n^2=^{2n}C_n$
If $\sqrt {x}$ is an irrational number whose square is an integer and y is an integer, such that $0<y-\sqrt {x}<1$ , then
- For even positive integer n,
  If $(y+\sqrt {x})^n = p+f$ , where $p\in I$ and and $(y-\sqrt {x})^n=f_1$ , where
  Then and p is an even integer
  Also
- For odd positive integer n and $0<\sqrt {x}-y<1$
  if $(\sqrt {x}+y)^n=\rho +f$ , where $\rho \in I,0<f<1$ and $(\sqrt {x}-y)^n=f_1$ , where
  Then and p is an odd integer $(\rho +f)f= (x-y^2)^n$
Properties of : is also denoted by $\Big(\overset {n}{\underset {r}{}}\Big)$

(a) $^nC_{r-1}+^nC_r=^{n+1}C_r$
(b) $^nC_x=^nC_y\Rightarrow x=y \quad or \quad x+y=n$
© $^nC_r=^nC_{n-r}$
(d) $r.^nC_r=n.^{n-1}C_{r-1}$
(e) $^nC_r=\frac {n}{r}.^{n-1}C_{r-1}$
(f) $\frac {^nC_r}{r+1}= \frac {^{n+1}C_{r+1}}{n+1}$
(g) $\frac {^nC_r}{^nC_{r-1}}= \frac {n-r+1}{r}$
(h) $\frac {^nC_r}{^nC_{r+1}}= \frac {r+1}{n-r}$
(i) If n is even, is greatest when $r= \frac {n}{2}$ .
If n is odd, is greatest when $r=\frac {n+1}{2}or \frac {n-1}{2}$
- In any expansion containing x, term independent of x means coefficient of
  .
- Coeff. of in $x^k\Big(x-\frac {3}{x^2}\Big)^n= coeff.of \quad x^{r-k}in \Big(x- \frac {3}{x^2}\Big)^n$
- may be taken as $a_0+a_1x+a_2x^2+\cdots +a_{4n}x^{4n}$ , where some of $a_0,a_1,\cdots ,a_{4n}$ , may be zero.
Multinomial theorem:
The expansion of $(x_1+x_2+x_3+x_4+\cdots +x_m)^n$
when $n \in N$ can be obtained by means of the binomial theorem.
The general term in the expansion of $(x_1+x_2+x_3+x_4+\cdots +x_m)^n$ $= \frac {n}{\alpha_1\alpha_2\cdots \alpha_m}x_1 \quad ^{\alpha 1}x_2 \quad ^{\alpha 2}\cdots x_m \quad ^{\alpha m}$
where $\alpha_1,\alpha_2,\cdots ,\alpha_m$ are nonâ€"negative integers satisfying the condition $\alpha_1+\alpha_2+\cdots +\alpha_m=n$

Proof: General term in the expansion $[x_1+(x_2+\cdots +x_m)]^n$
$=\frac {n}{\alpha_1|n - \alpha_1}x_1 \quad \alpha_1(x_2+x_3+\cdots +x_m)^{n-\alpha_1}\qquad \qquad \cdots (1)$
Again general term in the expansion of $(x_2+x_3+\cdots +x_m)^{n-\alpha 1}$ $=\frac {n-\alpha_1}{\alpha_2 |n-\alpha_1-\alpha_2}x_2\alpha_2.(x_3+x_4+\cdots +x_m)^{n-\alpha_1-\alpha_2} \qquad \cdots (2)$
Similarly general term in the expansion of $(x_3+x_4+\cdots +x_m)^{n-\alpha_1-\alpha_2}=$ $\frac {n-\alpha_1-\alpha_2}{\alpha_3|n-\alpha_1-\alpha_2-\alpha_3}x_3^{\alpha 3}$
$(x_4+x_5+\cdots x_m)^{n-\alpha_1-\alpha_2-\alpha_3} \qquad \qquad \qquad \qquad \qquad \cdots (3)$
Proceeding in this way we can show that the general term in the expansion of $(x_1+x_2+\cdots +x_m)^n$

$=\frac {n}{\alpha_1|n-\alpha_1}.$ $\frac {|n-\alpha_1}{|\alpha_2|n-\alpha_1-\alpha_2}$ $.\frac {|n-\alpha_1-\alpha_2}{|\alpha_3|n-\alpha_1-\alpha_2-\alpha_3}$ $.x_1^{\alpha_1}x_2^{\alpha_2}.$ $x_3^{\alpha_3}\cdots$ $x_m \quad ^{\alpha_m}$

where $\alpha_1+\alpha_2+\cdots +\alpha_m=n$ and $\alpha_1,\alpha_2,\cdots ,\alpha_m$ are nonâ€"negative integers.

(a) General term in $(a+b+c+d)^n=\frac {|n}{|p|q|r|s}a^p.b^q.c^r.d^s$ , where $p+q+r+s=n;p,q,r,s,\in[\{0,1,2,\cdots ,n\}]$
(b) Number of terms in the expansion of $(x_1+x_2+\cdots +x_m)^n$ = number of ways of distributing n identical things among m persons when each person can get zero or more things $=m\quad + \quad n \quad - \quad ^1C_n \quad or \quad m \quad + \quad n \quad - \quad ^1C_m \quad - \quad 1$