If for , then m equals: [2026]

Question

If for $3 \leq r \leq 30, ({}^{30}C_{30 - r}) + 3 ({}^{30}C_{31 - r}) + 3 ({}^{30}C_{32 - r}) + ({}^{30}C_{33 - r}) = {}^{m}C_{r + 1}$ , then m equals: [2026]

Smart Achievers · Accepted Answer

(3)

Since, ${}^{30}C_{30 - r} = {}^{30}C_{r}$ , ${}^{30}C_{31 - r} = {}^{30}C_{r - 1}$ , ${}^{30}C_{32 - r} = {}^{30}C_{r - 2}$ , and ${}^{30}C_{33 - r} = {}^{30}C_{r - 3}$

$∴ {}^{30}C_{30 - r} + 3 ({}^{30}C_{31 - r}) + 3 ({}^{30}C_{32 - r}) + {}^{30}C_{33 - r} = {}^{m}C_{r}$

$\Rightarrow {}^{30}C_{r} + 3 [{}^{30}C_{r - 1} + {}^{30}C_{r - 2}] + {}^{30}C_{r - 3} = {}^{m}C_{r}$

$\Rightarrow ({}^{30}C_{r} + {}^{30}C_{r - 1}) + 2 ({}^{30}C_{r - 1} + {}^{30}C_{r - 2}) + ({}^{30}C_{r - 2} + {}^{30}C_{r - 3}) = {}^{m}C_{r}$

$\Rightarrow {}^{31}C_{r} + 2 \cdot ({}^{31}C_{r - 1}) + {}^{31}C_{r - 2} = {}^{m}C_{r}$

$\Rightarrow {}^{31}C_{r} + {}^{31}C_{r - 1} + {}^{31}C_{r - 1} + {}^{32}C_{r - 2} = {}^{m}C_{r}$

$\Rightarrow {}^{32}C_{r} + {}^{32}C_{r - 1} = {}^{m}C_{r} \Rightarrow + {}^{33}C_{r} = {}^{m}C_{r}$

$\Rightarrow m = 33$

Solution

Q.
If for $3 \leq r \leq 30, ({}^{30}C_{30 - r}) + 3 ({}^{30}C_{31 - r}) + 3 ({}^{30}C_{32 - r}) + ({}^{30}C_{33 - r}) = {}^{m}C_{r + 1}$ , then m equals: [2026]

1 31

2 32

3 33

4 34

Want Us to Email you About Special Offers & Updates?