Final answer:
In 1,4-diphenyl-1,3-butadiene, there are two pi bonds within the butadiene core. The phenyl groups each have additional pi bonds, but these are not counted in the context of the question.
Step-by-step explanation:
The number of pi bonds in 1,4-diphenyl-1,3-butadiene can be determined by analyzing its structure. Butadiene is a 4-carbon molecule with two double bonds, and each double bond comprises one sigma (σ) bond and one pi (π) bond. Hence, in butadiene, there are two pi bonds. Since 1,4-diphenyl-1,3-butadiene is simply butadiene with a phenyl group attached to the first and fourth carbons, the number of pi bonds in the butadiene core remains the same, that is, two pi bonds.
If we consider the phenyl groups (which are aromatic rings), each has three additional pi bonds, but they are not usually counted when considering the pi bond content of conjugated systems such as 1,4-diphenyl-1,3-butadiene. Thus, the answer to the question is that 1,4-diphenyl-1,3-butadiene has two pi bonds in its conjugated system.