Problem Statement

Find the sum of all multinomial coefficients:

\[\binom{7}{b_1, b_2, b_3, b_4}\]

where \(b_1 + b_2 + b_3 + b_4 = 7\) and each \(b_i \geq 0\) is an integer.

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Solution

Step 1: Understanding the Multinomial Coefficients

Each multinomial coefficient

\[\binom{7}{b_1, b_2, b_3, b_4}\]

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:

\[\sum_{b_1 + b_2 + b_3 + b_4 = 7} \binom{7}{b_1, b_2, b_3, b_4}.\]

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 \(\mathbf{16384}\)

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