P(chosen at least once)

= P(chosen only once) + P(chosen twice)

= P(first name is Nicholas, but not the second)
+ P(second name is Nicholas, but not the first)
+ P(both names are Nicholas)

= (1/10 x 9/10) + (9/10 x 1/10) + (1/10 x 1/10)

= 9/100 + 9/100 + 1/100
= 19/100

Complement method :

P(Nicholas is chosen at least once)

= 1 - P(Nicholas is not chosen at all)

= 1 - (both first and second names are not Nicholas)

= 1 - (9/10 x 9/10)

= 1 - 81/100

= 19/100

The question, if you observe carefully, is similar to

"Ten balls are placed in a container. One of the balls is red. Two balls are drawn with replacement."

Yes same concept. Royal, please learn to identify similar questions and apply the same strategy. You'll need to draw parallels across different questions