Advertisements
Advertisements
प्रश्न
In the expansion of (1 + x)n the binomial coefficients of three consecutive terms are respectively 220, 495 and 792, find the value of n.
उत्तर
\[\text{ Suppose the three consecutive terms are } T_{r - 1} , T_r \text{ and } T_{r + 1} . \]
\[\text{ Coefficients of these terms are } ^{n}{}{C}_{r - 2} , ^{n}{}{C}_{r - 1} \text{ and } ^{n}{}{C}_r , respectively . \]
\[\text{ These coefficients are equal to 220, 495 and 792 } . \]
\[ \therefore \frac{^{n}{}{C}_{r - 2}}{^{n}{}{C}_{r - 1}} = \frac{220}{495}\]
\[ \Rightarrow \frac{r - 1}{n - r + 2} = \frac{4}{9}\]
\[ \Rightarrow 9r - 9 = 4n - 4r + 8\]
\[ \Rightarrow 4n + 17 = 13r . . . \left( 1 \right)\]
\[\text{ Also } , \]
\[\frac{^ {n}{}{C}_r}{^ {n}{}{C}_{r - 1}} = \frac{792}{495}\]
\[ \Rightarrow \frac{n - r + 1}{r} = \frac{8}{5}\]
\[ \Rightarrow 5n - 5r + 5 = 8r\]
\[ \Rightarrow 5n + 5 = 13r\]
\[ \Rightarrow 5n + 5 = 4n + 17 \left[ \text{ From Eqn} \left( 1 \right) \right]\]
\[ \Rightarrow n = 12\]
APPEARS IN
संबंधित प्रश्न
Write the general term in the expansion of (x2 – y)6
Find the middle terms in the expansions of `(3 - x^3/6)^7`
Find the middle term in the expansion of:
(i) \[\left( \frac{2}{3}x - \frac{3}{2x} \right)^{20}\]
Find the middle term in the expansion of:
(iii) \[\left( x^2 - \frac{2}{x} \right)^{10}\]
Find the middle terms in the expansion of:
(i) \[\left( 3x - \frac{x^3}{6} \right)^9\]
Find the middle terms in the expansion of:
(ii) \[\left( 2 x^2 - \frac{1}{x} \right)^7\]
Find the middle terms in the expansion of:
(iv) \[\left( x^4 - \frac{1}{x^3} \right)^{11}\]
Find the middle terms(s) in the expansion of:
(ii) \[\left( 1 - 2x + x^2 \right)^n\]
Find the middle terms(s) in the expansion of:
(viii) \[\left( 2ax - \frac{b}{x^2} \right)^{12}\]
Find the middle terms(s) in the expansion of:
(ix) \[\left( \frac{p}{x} + \frac{x}{p} \right)^9\]
Find the term independent of x in the expansion of the expression:
(iii) \[\left( 2 x^2 - \frac{3}{x^3} \right)^{25}\]
Find the term independent of x in the expansion of the expression:
(vi) \[\left( x - \frac{1}{x^2} \right)^{3n}\]
Find the term independent of x in the expansion of the expression:
(vii) \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\]
Prove that the coefficient of (r + 1)th term in the expansion of (1 + x)n + 1 is equal to the sum of the coefficients of rth and (r + 1)th terms in the expansion of (1 + x)n.
Find the coefficient of a4 in the product (1 + 2a)4 (2 − a)5 using binomial theorem.
If the 6th, 7th and 8th terms in the expansion of (x + a)n are respectively 112, 7 and 1/4, find x, a, n.
If p is a real number and if the middle term in the expansion of \[\left( \frac{p}{2} + 2 \right)^8\] is 1120, find p.
Write the total number of terms in the expansion of \[\left( x + a \right)^{100} + \left( x - a \right)^{100}\] .
If in the expansion of (a + b)n and (a + b)n + 3, the ratio of the coefficients of second and third terms, and third and fourth terms respectively are equal, then n is
If A and B are the sums of odd and even terms respectively in the expansion of (x + a)n, then (x + a)2n − (x − a)2n is equal to
If in the expansion of (1 + y)n, the coefficients of 5th, 6th and 7th terms are in A.P., then nis equal to
In the expansion of \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\] , the term independent of x is
If the sum of odd numbered terms and the sum of even numbered terms in the expansion of \[\left( x + a \right)^n\] are A and B respectively, then the value of \[\left( x^2 - a^2 \right)^n\] is
The number of terms with integral coefficients in the expansion of \[\left( {17}^{1/3} + {35}^{1/2} x \right)^{600}\] is
Find numerically the greatest term in the expansion of (2 + 3x)9, where x = `3/2`.
If xp occurs in the expansion of `(x^2 + 1/x)^(2n)`, prove that its coefficient is `(2n!)/(((4n - p)/3)!((2n + p)/3)!)`
Find the term independent of x in the expansion of (1 + x + 2x3) `(3/2 x^2 - 1/(3x))^9`
If the middle term of `(1/x + x sin x)^10` is equal to `7 7/8`, then value of x is ______.
In the expansion of `(x^2 - 1/x^2)^16`, the value of constant term is ______.
Middle term in the expansion of (a3 + ba)28 is ______.
The sum of coefficients of the two middle terms in the expansion of (1 + x)2n–1 is equal to 2n–1Cn.
The number of rational terms in the binomial expansion of `(4^(1/4) + 5^(1/6))^120` is ______.
Let the coefficients of the middle terms in the expansion of `(1/sqrt(6) + βx)^4, (1 - 3βx)^2` and `(1 - β/2x)^6, β > 0`, common difference of this A.P., then `50 - (2d)/β^2` is equal to ______.
