Question

A cupboard contains 55 pairs of​ shoes, each of a different style. How many ways are there to select 44 of the shoes from the cupboard so that the selection contains exactly one matching​ pair? Note that each pair of shoes consists of a distinguishable left and right foot.

Answer 1

1) General choice of shoes: 

C(6,1) = choose the matching pair = 6 choices. 
C(10,1) = 1 shoe from the other 10. 
Now you cannot choose the mate of that one, 
so you have 8 other shoes to choose from. 

6 * 10 * 8 = 480 choices of "4 shoes". 
But then we must divide by 2, because the last two shoes 
could be chosen in either order. 

That gives 240 choices