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I was looking for a combinatorical explenation for this identity $$\sum_{x_1+x_2+x_3 \le18}\binom{18}{x_1,x_2,x_3} = 4^{18}$$ A simple explenation would be enough,

Thanks

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This just the multinomial theorem for $(1+a+b+c)^{18}$ where $a = b = c = 1$.

Just leave a comment, if you need a less simple explanation :-)

Edit: Please note, that at Wikipedia's page there's a bit different definition of multinomial than yours, i.e. $\binom{n}{k_1, k_2, k_3}$ as compared to $\binom{n}{k_1, k_2, k_3, k_4}$ where $k_4 = n-k_1-k_2-k_3$.

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