Given 90% of the people are right handed.

A person can be right handed or left handed.

Probability of right handed person p = 90/100 = 9/10

Probability of left handed person q = 1 - 9/10 = 1/10

Now using Binomial theorem,

probability that more than 6 people are right handed = ∑ ¹⁰C_r * p^r * q^10-r                     (7<=r<=10)

                                                                                  = ∑ ¹⁰Cr * (9/10)^r * (1/10)^10-r

                                                                                  = ∑ ¹⁰C_r * (0.9)^r * (0.1)^10-r   

So the probability that at most 6 people are right handed = 1- {∑ ¹⁰C_r * (0.9)^r * (0.1)^10-r  }               (7<=r<=10)