NIMCET 2020 — Mathematics PYQ

1. Identify the given information:

• Total number of friends ($n$) = $10$

• Number of friends to be invited ($r$) = $6$

• Number of particular friends who must never be invited = $3$

2. Adjust the pool of friends:

Since those 3 particular friends cannot be chosen, we remove them from the total group of 10.

$$\text{Remaining friends to choose from}=10-3=7$$

3. Apply the combination formula:

Now, we need to choose 6 guests out of the remaining 7 friends. The number of ways to do this is given by the combination formula ${}_n{C}_r$ or $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$:

$$\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}$$

Substituting our values ($n=7$ and $r=6$):

$$\left(\genfrac{}{}{0ex}{}{7}{6}\right)=\frac{7!}{6!(7-6)!}$$

$$\left(\genfrac{}{}{0ex}{}{7}{6}\right)=\frac{7\times 6!}{6!\times 1!}$$

$$\left(\genfrac{}{}{0ex}{}{7}{6}\right)=7$$

Conclusion

There are exactly 7 times (or combinations) where those 3 particular friends will never attend the party.

Therefore, the correct option is B) 7.