A crew of an eight oar boat has to be chosen out of $11$ men five of whom can row on stroke side only, four on the bow side only, and the remaining two on either side. How many different selections can be made?
My attempt is as follows:-
Assuming there will be $4$ oars on stroke side and $4$ oars on bow side
Case $1$: Not selecting men who can row on both sides $\implies {5\choose 4}{4\choose 4}=5$
Case $2$: Only selecting one men out of $2$ for stroke side $\implies {2\choose 1}{5\choose 3}{4\choose 4}=20$
Case $3$: Only selecting one men out of $2$ for bow side $\implies {5\choose 4}{2\choose 1}{4\choose 3}=40$
Case $4$: Selecting both men for stroke side $\implies {5\choose 2}{4\choose 4}=10$
Case $5$: Selecting both men for bow side $\implies {5\choose 4}{4\choose 2}=30$
So I am getting total $5+20+40+10+30=105$ ways, but actual answer $145$. What mistake am I making here?
Note: In all above cases, selection of men we are talking about are the ones who can row on both sides.
Case $4\frac{1}{2}$: Selecting both men for one for stroke side & one for bow side $\implies {5\choose 3}{4\choose 3}=40 $.