The number of ordered pairs (r, k) for which | vedclass

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

Question

(C) Given the equation: $6 \cdot ^{35}C_{r} = (k^{2} - 3) \cdot ^{36}C_{r+1}$.Using the identity $^{n}C_{r} = \frac{n}{r} \cdot ^{n-1}C_{r-1}$ or $^{n

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(C) Given the equation: $6 \cdot ^{35}C_{r} = (k^{2} - 3) \cdot ^{36}C_{r+1}$.Using the identity $^{n}C_{r} = \frac{n}{r} \cdot ^{n-1}C_{r-1}$ or $^{n+1}C_{r+1} = \frac{n+1}{r+1} \cdot ^{n}C_{r}$,we have $^{36}C_{r+1} = \frac{36}{r+1} \cdot ^{35}C_{r}$.Substituting this into the equation:$6 \cdot ^{35}C_{r} = (k^{2} - 3) \cdot \frac{36}{r+1} \cdot ^{35}C_{r}$.Since $^{35}C_{r} 
eq 0$,we can divide both sides by $^{35}C_{r}$:$6 = (k^{2} - 3) \cdot \frac{36}{r+1}$.Rearranging for $(k^{2} - 3)$:$k^{2} - 3 = \frac{6(r+1)}{36} = \frac{r+1}{6}$.Thus,$k^{2} = \frac{r+1}{6} + 3 = \frac{r+19}{6}$.Since $k$ is an integer,$k^{2}$ must be a perfect square. Also,$0 \le r \le 35$.For $r=5$,$k^{2} = \frac{5+19}{6} = 4 \Rightarrow k = \pm 2$.For $r=35$,$k^{2} = \frac{35+19}{6} = 9 \Rightarrow k = \pm 3$.These give the ordered pairs $(5, 2), (5, -2), (35, 3), (35, -3)$.There are $4$ such ordered pairs.

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

Similar Questions

$A$ committee of $7$ members is to be formed from $9$ boys and $4$ girls. In how many ways can this be done if the committee must consist of at least $3$ girls?

The number of non-negative integral solutions of $x_1+x_2+x_3+x_4=10$ is

$A$ committee of $4$ persons is to be formed from $2$ ladies,$2$ old men,and $4$ young men such that it includes at least $1$ lady,at least $1$ old man,and at most $2$ young men. The total number of ways in which this committee can be formed is:

In how many ways can a group of $7$ persons be formed from $6$ boys and $4$ girls such that the boys are in the majority?

Out of $14$ cricket players,$5$ are bowlers. In how many ways can a team of $11$ players be selected such that the team contains at least $4$ bowlers?

Vedclass Test Series

Exam Paper Generator

Online Exam Module