In this paper, we prove that the multicolored Ramsey number $R(G_1,\dots,G_n,K_{n_1},\dots,K_{n_r})$ is at least $(γ-1)(κ-1)+1$ for arbitrary connected graphs $G_1,\dots,G_n$ and $n_1,\dots,n_r\in\mathbb{N}$, where $γ=R(G_1,\dots,G_n)$ and $κ=R(K_{n_1},\dots,K_{n_r})$. Erd\H os at al. conjectured that $R(C_n,K_l)=(n-1)(l-1)+1$ for every $n\geq l\geq 3$ except for $n=l=3$. Nikiforov proved this conjecture for $n\geq 4l+2$. Using the above bound, we derive the following generalization of this result. $R(C_n,K_{n_1},\dots,K_{n_r})=(n-1)(κ-1)+1$, where $κ=R(K_{n_1},\dots,K_{n_r})$ and $n\geq 4κ+2$.