Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 20199 AprilEvening Shift
Exam
JEE Main
Year
2019
Date
9 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2019 · 9 April · Evening Shift · Mathematics · Binomial Theorem

If some three consecutive coefficients in the binomial expansion of...

If some three consecutive coefficients in the binomial expansion of (x+1)ninpowersof{\left(x + 1\right)}^{n} in powers ofxareintheratioare in the ratio2: 15: 70, then the average of these three coefficients is:

Options

  1. A227227
  2. B964964
  3. C625625
  4. D232232

Correct answer

D. 232232

Step-by-step solution

From the given condition

nCr:nCr+1:nCr+2=2:15:70{}^{n}{C}_{r}{:}^{n}{C}_{r + 1}{:}^{n}{C}_{r + 2} = 2: 15: 70

⇒nCrnCr+1=215⇒ \dfrac{{}^{n}{C}_{r}}{{}^{n}{C}_{r + 1}} = \dfrac{2}{15} and nCr+1nCr+2=1570\dfrac{{}^{n}{C}_{r + 1}}{{}^{n}{C}_{r + 2}} = \dfrac{15}{70}

⇒(n!(n−r)!⋅r!)(n!(n−r−1)!⋅(r+1)!)=215⇒ \dfrac{\left(\dfrac{n !}{\left(n - r\right) ! \cdot r !}\right)}{\left(\dfrac{n !}{\left(n - r - 1\right) ! \cdot \left(r + 1\right) !}\right)} = \dfrac{2}{15} and (n!(n−r−1)!⋅(r+1)!)(n!(n−r−2)!⋅(r+2)!)=1570\dfrac{\left(\dfrac{n !}{\left(n - r - 1\right) ! \cdot \left(r + 1\right) !}\right)}{\left(\dfrac{n !}{\left(n - r - 2\right) ! \cdot \left(r + 2\right) !}\right)} = \dfrac{15}{70}

⇒(n−r−1)!⋅(r+1)!(n−r)!⋅r!=215⇒ \dfrac{\left(n - r - 1\right) ! \cdot \left(r + 1\right) !}{\left(n - r\right) ! \cdot r !} = \dfrac{2}{15} and (n−r−2)!⋅(r+2)!(n−r−1)!⋅(r+1)!=314\dfrac{\left(n - r - 2\right) ! \cdot \left(r + 2\right) !}{\left(n - r - 1\right) ! \cdot \left(r + 1\right) !} = \dfrac{3}{14}

⇒(n−r−1)!⋅(r+1)⋅r!(n−r)⋅(n−r−1)!⋅r!=215⇒ \dfrac{\left(n - r - 1\right) ! \cdot \left(r + 1\right) \cdot r !}{\left(n - r\right) \cdot \left(n - r - 1\right) ! \cdot r !} = \dfrac{2}{15} and (n−r−2)!⋅(r+2)⋅(r+1)!(n−r−1)⋅(n−r−2)!⋅(r+1)!=314\dfrac{\left(n - r - 2\right) ! \cdot \left(r + 2\right) \cdot \left(r + 1\right) !}{\left(n - r - 1\right) \cdot \left(n - r - 2\right) ! \cdot \left(r + 1\right) !} = \dfrac{3}{14}

⇒r+1n−r=215⇒ \dfrac{r + 1}{n - r} = \dfrac{2}{15} and r+2n−r−1=314\dfrac{r + 2}{n - r - 1} = \dfrac{3}{14}

⇒17r=2n−15⇒ 17r = 2n - 15 and 17r=3n−3117r = 3n - 31

⇒3n−31=2n−15,⇒n=16⇒ 3n - 31 = 2n - 15, ⇒ n = 16 and r=1r = 1

Hence, average =nCr+nCr+1+nCr+23= \dfrac{{}^{n}{C}_{r}{+}^{n}{C}_{r + 1}{+}^{n}{C}_{r + 2}}{3}

=16C1+16C2+16C33= \dfrac{{}^{16}{C}_{1}{+}^{16}{C}_{2}{+}^{16}{C}_{3}}{3}

=232.= 232 .

Practice Binomial Theorem on Quantrex Academy →

More from Binomial Theorem

If 26((2^3)/(3) 12 2 + (2^5)/(5) 12 4 + (2^7)/(7) 12 6 + … + 2^ 13 13 12 12 ) = 3^ 13 - α… JEE Main 2026If (1 - x^3)^ 10 = ∑ _ r=0 ^ 10 a_r x^r (1-x)^ 30-2r , then 9a_9 a_ 10 is equal to… JEE Main 2026If the coefficients of the middle terms in the binomial expansions of (1 + α x)^ 26 and… JEE Main 2026The coefficient of x^2 in the expansion of (2x^2 + (1)/(x))^ 10 , x ≠ 0 , is : JEE Main 2026If the sum of the coefficients of x^7 and x^ 14 in the expansion of ((1)/(x^3) - x^4)^n… JEE Main 2026In the expansion of (9x- 1 3 x )^ 18 , x>0 , if the term independent of x is (221)k… JEE Main 2026Let the smallest value of k ∈ ℕ , for which the coefficient of x^3 in (1+x)^3 + (1+x)^4 +… JEE Main 2026If for 3 ≤ r ≤ 30 , 30 30-r + 3 30 31-r + 3 30 32-r + 30 33-r = m r , then m equals: JEE Main 2026The number of elements in the set S = \ (r, k) : k ∈ ℤ and ^ 36 C_ r+1 = 6( ^ 35 C_r)… JEE Main 2026Given below are two statements : Statement I: 25^ 13 +20^ 13 +8^ 13 +3^ 13 is divisible… JEE Main 2026The sum of the coefficients of x^ 499 and x^ 500 in (1+x)^ 1000 +x(1+x)^ 999 +x^ 2 (1+x)^… JEE Main 2026Let S=(1)/(25!)+(1)/(3!23!)+(1)/(5!21!)+… up to 13 terms. If 13 ~S= 2^ k n! , k ∈ ~N… JEE Main 2026The value of ^ 100 C_ 50 51 + ^ 100 C_ 51 52 +….+ ^ 100 C_ 100 101 is : JEE Main 2026The sum of all possible values of n ∈ N , so that the coefficients of x, x^ 2 and x^ 3 in… JEE Main 2026Let C_ r denote the coefficient of x^ r in the binomial expansion of (1+x)^ n , n ∈ N, 0… JEE Main 2026The coefficient of x^ 48 in (1+x)+2(1+x)^ 2 +3(1+x)^ 3 +…+100(1+x)^ 100 is equal to JEE Main 2026 Full Binomial Theorem list Binomial Theorem JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs