A soccer ball is made with 32 pieces of leather. Of these pieces, 12 are identical regular pentagons and the other 20 are also regular and identical hexagons. The sides of the pentagons are equal to the sides of the hexagons. To join two sides of two of these pieces, a seam is required. How many seams are needed to make a ball?
(a) 60
(b) 64
(c) 90
(d) 120
(e) 180
The correct option is (c).

If we add up the number of sides of all the polygons (20 hexagons and 12 pentagons) that make up the surface of the ball, we will get a value that is twice the number of seams, since each seam is a side common to exactly two polygons. Thus, we have $2 \times$ number of seams $=12 \times 5+20 \times 6=180$, from which the number of seams is 90.