The least value of natural number n satisfyin | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Using the Pascal's identity,$^nC_r + ^nC_{r-1} = ^{n+1}C_r$,we have $^nC_6 + ^nC_5 = ^{n+1}C_6$.Given the inequality: $^{n+1}C_6 > ^{n+1}C_5$.Expa

Vedclass · Accepted Answer

$11$

Vedclass · Answer

(A) Using the Pascal's identity,$^nC_r + ^nC_{r-1} = ^{n+1}C_r$,we have $^nC_6 + ^nC_5 = ^{n+1}C_6$.Given the inequality: $^{n+1}C_6 > ^{n+1}C_5$.Expanding the combinations: $\frac{(n+1)!}{6!(n-5)!} > \frac{(n+1)!}{5!(n-4)!}$.Dividing both sides by $(n+1)!$ and simplifying the factorials:$\frac{1}{6 	imes 5!(n-5)!} > \frac{1}{5!(n-4)(n-5)!}$.$\frac{1}{6} > \frac{1}{n-4}$.Since $n$ is a natural number,$n-4 > 6$,which implies $n > 10$.The least natural number $n$ satisfying this condition is $11$.

The least value of natural number $n$ satisfying $C(n, 5) + C(n, 6) > C(n + 1, 5)$ is

Similar Questions

In how many different ways can the letters of the word '$CYCLE$' be arranged?

It is required to seat $5$ men and $4$ women in a row so that the women occupy the even places. How many such arrangements are possible?

The number of groups that can be made from $5$ different green balls,$4$ different blue balls,and $3$ different red balls,if at least $1$ green and $1$ blue ball is to be included.

The number of words that do not start and end with vowels,formed using all the letters of the word $UNIVERSITY$ such that all vowels are in alphabetical order,is

The number of ordered pairs $(r, k)$ for which $6 \cdot ^{35} C_{r} = (k^2 - 3) \cdot ^{36} C_{r+1}$,where $k$ is an integer,is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module