Question

For the given equation ${}^{10}{C}_r+2\cdot {}^{10}{C}_{r-1}+{}^{10}{C}_{r-2}={}^{12}{C}_s$, which of the following is/are correct?

• A. If $r=s$, then the possible number of solutions is $13$
• B. If $r\ne s$, then the possible number of solutions is $12$
• C. If $r=s$, then the possible number of solutions is $9$
• D. If $r\ne s$, then the possible number of solutions is $8$

Answer 1

Answer: (C), (D)

If $r\ne s$, then the possible number of solutions is $8$

Given : ${}^{10}{C}_r+2\cdot {}^{10}{C}_{r-1}+{}^{10}{C}_{r-2}={}^{12}{C}_s$
For combination to be defined
$10\ge r\ge 2$
Now,
${}^{10}{C}_r+{}^{10}{C}_{r-1}+{}^{10}{C}_{r-1}+{}^{10}{C}_{r-2}={}^{11}{C}_r+{}^{11}{C}_{r-1}={}^{12}{C}_r$
Therefore,
${}^{12}{C}_r={}^{12}{C}_s$
When $r=s$, then
$r=2,3,4,5,6,7,8,9,10$
The number of solutions is $9$

When $r+s=12$, then
$(r,s)=(2,10),(3,9),(4,8),(5,7),(7,5),(8,4),(9,3),(10,2)$
The number of solutions is $8$