Combinatorial Probability

I'm experiencing difficulty solving one of the exercises in my probability and statistics. Any help would be appreciated.

**Exercise 1-29** At a picnic, there was a bowl of chocolate candy that had 10 pieces each of Milky Way, Almond Joy, Butterfinger, Nestle Crunch, Snickers, and Kit Kat. Jen grabbed six pieces at random from this bowl of 60 chocolate candies.

(b) What is the probability that Jen grabs exactly five varieties?

I know there are $\displaystyle {60\choose6}$ combinations of grabbing 6 pieces of candy at random from the bowl of 60 chocolate candies. I believe there are $\displaystyle {10\choose1} \times {10\choose1} \times {10\choose1} \times {10\choose1} \times {10\choose1} \times {45\choose1}$ outcomes in which Jen grabs exactly five varieties. Therefore, the probability that Jen grabs exactly five varieties is approximately 0.089885. However, the solutions manual gives the answer as 0.05393.