The number of elements in the set , is: [2026]

Question

The number of elements in the set $S = {(r, k) : k \in Z and {}^{36}C_{r + 1} = \frac{6 ({}^{35}C_{r})}{(k^{2} - 3)}}$ , is: [2026]

Smart Achievers · Accepted Answer

(2)

${}^{36}C_{r + 1} = \frac{6 {}^{35}C_{r}}{(k^{2} - 3)}$

$\Rightarrow \frac{{}^{36}C_{r + 1}}{{}^{35}C_{r}} = \frac{6}{k^{2} - 3}$

$\Rightarrow \frac{\frac{36!}{(r + 1)! \times (35 - r)!}}{\frac{35!}{r! (35 - r)!}} = \frac{6}{k^{2} - 3}$

$\Rightarrow \frac{36}{r + 1} = \frac{6}{k^{2} - 3} \Rightarrow 6 (k^{2} - 3) = r + 1$

$\Rightarrow 6 k^{2} - r = 19 \Rightarrow r = 6 k^{2} - 19$

Now, ${}^{35}C_{r}$ is defined, so $0 \leq 6 k^{2} \leq 35$

$\Rightarrow 0 \leq k^{2} - 19 \leq 35 \Rightarrow 19 \leq 6 k^{2} \leq 54$

$\Rightarrow \frac{19}{6} \leq 6 k^{2} \leq 9$

$\Rightarrow 3.1 \leq k^{2} \leq 9 \Rightarrow k \in {- 3, - 2, 2, 3}$

$\Rightarrow r \in {35, 5}$

So, (r, k) = {(35, –3), (35,3), (5, –2), (5, 2)}

Number of elements = 4.

Want Us to Email you About Special Offers & Updates?