Question:
If ${ }^{35} C_{n+7}={ }^{35} C_{4 n-2}$ then find the value of $n$.
Solution:
Given: ${ }^{35} \mathrm{C}_{n+7}={ }^{35} \mathrm{C}_{4 n-2}$ Need to find: Value of $n$ We know, one of the property
of combination is: If ${ }^{n} C_{r}={ }^{n} C_{t}$, then, (i) $r=t \mathrm{OR}$ (ii) $r+t=n$ Applying property (i) we get, $\Rightarrow \mathrm{n}+7=4 \mathrm{n}-2 \Rightarrow 3 \mathrm{n}=9 \Rightarrow \mathrm{n}=3$ Applying property (ii) we get, $\Rightarrow \mathrm{n}+7+4 \mathrm{n}-2=$ $35 \Rightarrow 5 \mathrm{n}=30 \Rightarrow \mathrm{n}=6$ Therefore, the value of $\mathrm{n}$ is either 3 or 6 .