Multinomial Sum

Step 1: Understanding the Multinomial Coefficients

Each multinomial coefficient

$$\left(\genfrac{}{}{0ex}{}{7}{b_1,b_2,b_3,b_4}\right)$$

represents the number of ways to assign the exponents $b_1,b_2,b_3,$ and $b_4$ in the expansion of:

$$(x_1+x_2+x_3+x_4{)}^7.$$

Thus, the sum of all such coefficients is simply the sum of all terms in this multinomial expansion when each $x_i=1$.

Step 2: Reduced Version - The Core Idea

We recognize that the given sum is:

$$\sum _{b_1+b_2+b_3+b_4=7}\left(\genfrac{}{}{0ex}{}{7}{b_1,b_2,b_3,b_4}\right).$$

Setting $x_1=x_2=x_3=x_4=1$, the expansion simplifies to:

$$(1+1+1+1{)}^7=4^7.$$

which directly gives us the sum.

Step 3: Compute the Final Value

$$4^7=16384.$$

Thus, the sum of all multinomial coefficients is $16384$