# Properties of Binomial Coefficients and Simple Applications

Tips :

•  nCo , nC1 , nC2 ......... nCn are called the binomial coefficients in the expansion of (x + a)n. They are denoted by Co , C1 , C2 .......... Cn
• Co , C1 , C2 , C3 ......... Cn are the binomial coefficients in the expansion of (1 + x)n
• nCo + nC1 + nC2 + ......... + nCn = 2n
• nCo - nC1 + nC2 - nC3 + ......... +(−1)n . nCn = 0
• nCo + nC2 + nC4 +........= nC1 + nC3 + .......... = 2n−1
• \tt C_{O} + \frac{C_{1}}{2}x + \frac{C_{2}}{3}x^{2}+......+\frac{C_{n}}{n+1}x^{n}=\frac{\left(1+x\right)^{n+1}-1}{\left(n+1\right)x}
• \tt C_{O} + \frac{C_{1}}{2}+\frac{C_{2}}{3}+\frac{C_{3}}{4}+..........+\frac{C_{n}}{n+1}=\frac{2^{n+1}-1}{n+1}
• \tt C_{O} - \frac{C_{1}}{2}+\frac{C_{2}}{3}-\frac{C_{3}}{4}+...........+\left(-1\right)^n\frac{C_{n}}{n+1}=\frac{1}{n+1}
• \tt C_{O} +\frac{C_{2}}{3}+\frac{C_{4}}{5}+........=\frac{2^n}{n+1}
• \tt \frac{C_{1}}{2}+\frac{C_{3}}{4}+\frac{C_{5}}{6}+.......=\frac{2{^n}-1}{n+1}
• a. Co + (a + d) C1 + (a + 2d) C2 + .........+ (a + nd). Cn = (2a + nd) 2n−1
• a. Co − (a + d) C1 + (a + 2d) C2 − .......... + (−1)n (a + nd). Cn = O
• \tt C_0^2 +C_1^2+C_2^2 +..........+C_n^2 = ^{2n}C_n
• \tt C_0^2 -C_1^2+C_2^2-C_3^2+ ........ +\left(-1\right)^{n}.C_n^2 = \left(-1\right)^{n/2}\ ^nC_\frac{n}{2}\left[n\ is\ even\right]=0,\left(n\ i \ odd\right)
• \tt a C_0^2 \left(a+d\right)C_1^2+\left(a+2d\right)C_2^2+ ...... +\left(a+nd\right)C_n^2=\left[\frac{2a+nd}{2}\right].^{2n}C_n
• Co. Cr + C1 Cr+1 + ...... +Cn−r. Cn = 2nCn−r

Tricks :

• The coefficient of xr in (1 + x)m . (1 + x)n is (m+n)Cr
• The coefficient of the term independent of ‘x’ in the expansion of \tt \left(1+x\right)^{m}\left(1+\frac{1}{x}\right)^{n} \ is ^{m+n}C_n
• If the greatest term in the expansion of (1 + x)2n has also greatest coefficient, then \tt \frac{n}{n+1}< x <\frac{n+1}{n}
• The greatest term in the expansion of (1 + x)2n-1 has also greatest coefficient, then \tt \frac{n-1}{n+1}< x <\frac{n+1}{n-1}
• The number of rational terms in the expansion of \tt \left[a^{\frac{1}{t}}+b^{\frac{1}{k}}\right]^{n} \ is \left[\frac{n}{L.C.M \ of \left\{l,k\right\}}\right] when none of ‘l’ and ‘k’ is a factor of ‘n’ and when atleast one of ‘l’ and k is a factor of ‘n’ is \tt \left[\frac{n}{L.C.M \ of \left\{l,k\right\}}\right]+1 where [·] greatest integer
• Let f(x) be any polynomial in x, sum of the coefficients = f(1)
• Sum of the coefficients of even powers of \tt x=\frac{f\left(1\right)+f\left(-1\right)}{2}
• Sum of the coefficients of odd powers of \tt x=\frac{f\left(1\right)-f\left(-1\right)}{2}
• Number of terms in (1 + x)n is “n + 1” when ‘n’ is positive integer infinite ‘n’ is not a positive integer and |x|<1
• First negative term in \tt \left(1+x\right)^{p/q} when 0 < x < 1 , p, q are positive integers and ‘P’ is not a multiple of ‘q’ is \tt \left[\frac{p}{q}\right]+3
• If ‘x’ is very small so that x2 and higher powers of ‘x’ can be neglected. Then
(1) \tt \left(1+x\right)^{\frac{p}{q}}=1+\frac{p}{q}x     (2) \tt \left(1+x\right)^{\frac{-p}{q}}=1-\frac{p}{q}x
(3) \tt \left(1-x\right)^{\frac{p}{q}}=1-\frac{p}{q}x       (4) \tt \left(1-x\right)^{\frac{-p}{q}}=1+\frac{p}{q}x
• The coefficient of xn−1 in the expansion of \tt \left(x-1\right)\left(x-2\right).......\left(x-n\right) \ is \ \frac{-n\left(n+1\right)}{2}
• The coefficient of xn−1 in the expansion of \tt \left(x+1\right)\left(x+2\right)......\left(x+n\right) \ is \ \frac{n\left(n+1\right)}{2}
• The coefficient of xn−2 in the expansion of \tt \left(x-1\right)\left(x-2\right)......\left(x-n\right) \ is \ \frac{n\left(n+1\right)\left(n-1\right)\left(3x+2\right)}{24}
• If the coefficients of pth, qth terms in the expansion of (1 + x)n are equal then p + q = n + 2
• If the coefficients of xr, xr+1 in the expansion of \tt \left(a+\frac{x}{b}\right)^{n} are equal, then n = (r + 1)(ab + 1) −1
• If nCr−1 : nCr : nCr+1 = a : b : c then \tt r=\frac{a\left(b+c\right)}{b^{2}-ac}, n=\frac{ab+bc+2ca}{b^{2}-ac}
• For n > 6 \tt \left(\frac{n}{3}\right)^{n}<n!<\left(\frac{n}{2}\right)^{n}
• nn+1 > (n + 1)n  ∀ n ∈ N
• nn > (n + 1)n-1  ∀ n ∈ N.

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Greatest Coefficient
a). If n is even, then in (x + a)n , the greatest coefficient is nCn/2.
b). If n is odd, then in (x + a)n , the greatest coefficient is \tt ^nC_\frac{n-1}{2} \ or \ ^nC_\frac{n+1}{2} both being equal.

2. If C0, C1, C2, .... Cn are the coefficients of (1 + x)n, then
i) nCr + nCr−1 = n+1Cr
ii) \frac{^nC_r}{^{n-1}C_{r-1}}=\frac{n}{r}
iii) \frac{^nC_r}{^{n}C_{r-1}}=\frac{n-r+1}{r}
iv) C0 + C1 + C2 + ...+ Cn = 2n
v) C0 + C2 + C4 + ... = C1 + C3 + C5 + .... = 2n−1
vi) C0 − C1 + C2 − C3 + ....+ (−1)n Cn = 0
vii) C0Cr + C1Cr+1 + ...+ Cn−r = 2nCn+r = \frac{\left(2n\right)!}{\left(n-r\right)!\left(n+r\right)!}
viii) C_0^2+C_1^2+C_2^2+...+C_n^2={^{2n}C_n}=\frac{\left(2n\right)!}{\left(n!\right)^{2}}
ix) C0 − C2 + C4 − C6 + ...= \left(\sqrt{2}\right)^{n}\cos \frac{n\pi}{4}
x) C1 − C3 + C5 − C7 + ...= \left(\sqrt{2}\right)^{n}\sin \frac{n\pi}{4}
xi) C0 − C1 + C2 − C3 + ... + (−1)r n−1 Cr, r > n
xii) C_0^2-C_1^2+C_2^2-C_3^2+ ... = \begin{cases}0,if \ n \ is \ odd. &\\\left(-1\right)^{n/2}\cdot {^nC_{n/2}}, if \ n \ is \ even \end{cases}
xiii) C_{0}+\frac{C_{1}}{2}+\frac{C_{2}}{3}+...+\frac{C_{n}}{n+1}=\frac{2^{n+1}-1}{\left(n+1\right)}
xiv) C_{0}-\frac{C_{1}}{2}+\frac{C_{2}}{3}-\frac{C_{3}}{4}+...+\left(-1\right)^{n}\frac{C_{n}}{n+1}=\frac{1}{n+1}
xv) C_{0}+\frac{C_{1}}{2}+\frac{C_{2}}{2^{2}}+\frac{C_{3}}{2^{3}}+...+\frac{C_{n}}{2^{n}}=\left(\frac{3}{2}\right)^{n}