Binomial Theorem
Average of binomial coefficients
nta_pyq_2025_apr
Grade 12
Question:
For an integer n$\ge$2, if the arithmetic mean of all coefficients in the binomial expansion of$(x + y)$$2n-3$is 16 , then the distance of the point P$(2n - 1$,$n - 4n)$from the line$x + y = 8$is: 2
$\sqrt{2}$
2$\sqrt{2}$
5$\sqrt{2}$
3$\sqrt{2}$n
Step-by-Step Solution
Key Concept: Use the general term of the binomial expansion for average of binomial coefficients and impose the required coefficient or exponent condition.
$(2n - 3 + 1)$No. of terms in $(x + y)$$(2n-3)$$\Rightarrow$ [ ]. (4)$(2n - 2)$∴ sum of all$coefficients = 2$$2n-3$(Put$x = y = 1$) ∴ Arithmetic mean of all coefficients$2n-3$$2 = ($$) = 16$$2n - 2$$2n-3$5 $\Rightarrow$$2 = 2$$(n - 1)$$\Rightarrow$$n = 5$2 ∴ P$(2n - 1$,$n - 4n) = (9$, 5)$x + y = 8$|$9 + 5 - 8$| 6 3 $\times$ 2 ∴ PM = | | = = = 3$\sqrt{2}$|$\sqrt{2}$|$\sqrt{2}$$\sqrt{2}$r
Correct Answer: 4