TITLE: Seating arrangements: Question about the book solution and summation indices
QUESTION [6 upvotes]: $20$ people are to be seated at seven tables, three of which have 4 seats and four of which have 2 seats. If the people are randomly seated, find the expected value of the number of married couples that are seated at the same table.

My question 1:
In the book solution (see screenshot below) I don't understand the last equality, specifically the indexes that go from $i=1$ to $22$ and $19$ respectively. It seems they should both go from $i=1$ to $10$.
My question 2:
Where am I going wrong in my approach?
Let $X$ be the number of married couples sitting at the same table. Let $X_i = 1$  if couple $i$ is sitting at the same table for $i=1,...,10$. Then
$$E[X] = E\left[\sum_{i=1}^{10} X_i \right] = \sum_{i=1}^{10} E\left[X_i \right] = 10 \cdot P(X_1 = 1)$$
where the last equality comes from LOE and symmetry. To find $P(X_1 = 1)$ I will condition on the event $A =$ the husband is at a table with $4$ seats and the event $B =$ the husband is at a table with $2$ seats.
$$P(X_1 = 1) = P(X_1 = 1 \mid A)P(A) + P(X_1 = 1 \mid B)P(B)$$
$$=\frac{3}{19}\frac{12}{20}+\frac{1}{19}\frac{8}{20} \approx .1157$$
and so $E[X] \approx 1.157$ which does not match the book solution of $2.48$. Where am I going off the rails?
Thanks for your help and patience.

Book solution

REPLY [5 votes]: You're right on all counts. The limits on the sums are wrong, and if you correct them, the result in the book coincides with yours (except the correct result rounds to $0.1158$, not $0.1157$).