1. We are given positive real numbers $a,b,c,d$ such that $a+b+c+d=11$ and want to find the value of $\beta$ if the maximum of $a^5 b^3 c^2 d$ is $3750 \beta$. 2. To maximize the product $a^5 b^3 c^2 d$ under the constraint $a+b+c+d=11$, we use the method of Lagrange multipliers or apply the weighted AM-GM inequality. 3. By the weighted AM-GM inequality, for positive $a,b,c,d$ and weights summing to 1, the maximum of $a^5 b^3 c^2 d$ occurs when $$\frac{a}{5} = \frac{b}{3} = \frac{c}{2} = d = t$$ for some positive $t$. 4. Since $a=5t$, $b=3t$, $c=2t$, and $d=t$, the sum constraint gives $$5t + 3t + 2t + t = 11t = 11 \implies t=1.$$ 5. Substitute back to find the maximum product: $$a^5 b^3 c^2 d = (5)^5 (3)^3 (2)^2 (1) = 5^5 \times 3^3 \times 2^2 = 3125 \times 27 \times 4.$$ 6. Calculate stepwise: $$3125 \times 27 = 84375,$$ $$84375 \times 4 = 337500.$$ 7. We have the maximum product as $337500 = 3750 \beta$, so $$\beta = \frac{337500}{3750} = 90.$$ 8. Therefore, the value of $\beta$ is 90. Final answer: $\boxed{90}$