Individuals are classified by income as lower$-$class $($L$),$ middle$-$class $($M$),$ or upper$-$class $($U$).$ It is asserted that the income class of a person is mostly determined by the income class of his parents. In an experiment, $1,000$ individuals were studied for their changes in income class from one generation to the next: $220$ were from $L$ class, $650$ from M$,$ and $130$ from U$.$ The individuals from L class had the following children: $327$ in L class, $137$ in M$,$ and $63$ in U$.$ The individuals from M class had $158$ children in L class, $756$ in M$,$ and $215$ in U$.$ The individuals from U class had $19$ children in L class, $59$ in M$,$ and $96$ in U$.$ Use Markov chains based analysis to answer the following questions.

c$)$ Based on the distribution of income classes for $1,000$ individuals from the original dataset $(22\%L,65\%M,13\%U),$ find the distribution of income classes for their grandchildren. $(10)$