Propositional Logic - Distributive Law - Help to Resolve My Conflict of Intuition

Question

I'm just starting to learn propositional logic, and I'm hung up on the part of the Distributive Law specifically on the distribution of the $\vee$ (or) operator, where if $P \vee (Q \wedge R)$ then $(P \vee Q) \wedge (P \vee R)$. My problem with the statement boils down to the fact that intuitively in the first clause, regardless of the truth value of $P$, the truth values of $Q$ and $R$ have to be the same as one another (both true or both false), right? However, in the second clause, my understanding intuitively is that if $P$ is true, then $Q$ and $R$ can have different truth values (and do not need to be both true or both false). Hence, I do not see how the Distributive Law equivalence holds, and I'm also having trouble finding where I'm going wrong. Thanks for your help.

Answer 1

Per @Z.A.K.,

'You ask: "in the first clause, regardless of the truth value of P, the truth values of Q and R have to be the same as one another, right?" Not right. E.g. it could be that P and Q are true, but R is false. Then Q and R have different truth values, Q∧R is not true, but P is true, and since one of the disjuncts is true, the whole disjunction P∨(Q∧R) is true.'