The value of $$(^{21}C_{1} - ^{10}C_{1}) + (^{21}C_{2} - ^{10}C_{2}) + \dots + (^{21}C_{10} - ^{10}C_{10})$$ is:

$$S = \underbrace{\sum_{k=1}^{10} {}^{21}C_{k}}_{S_1} - \underbrace{\sum_{k=1}^{10} {}^{10}C_{k}}_{S_2}$$

$$\sum_{k=0}^{10} {}^{10}C_{k} = 2^{10}$$

$$S_2 = \sum_{k=1}^{10} {}^{10}C_{k} = 2^{10} - {}^{10}C_{0} = \mathbf{2^{10} - 1}$$

For $$n = 21$$, the total sum is $$\sum_{k=0}^{21} {}^{21}C_{k} = 2^{21}$$

Since $${}^{n}C_{k} = {}^{n}C_{n-k}$$, the sum of the first half of the terms is equal to the second half:

$$\sum_{k=0}^{10} {}^{21}C_{k} = \sum_{k=11}^{21} {}^{21}C_{k} = \frac{2^{21}}{2} = 2^{20}$$

$$S_1 = \sum_{k=1}^{10} {}^{21}C_{k} = 2^{20} - {}^{21}C_{0} = \mathbf{2^{20} - 1}$$

$$S = (2^{20} - 1) - (2^{10} - 1)$$

$$S = 2^{20} - 2^{10}$$