Find the value of $\sum_{r=0}^{10} \binom{10}{r}\cdot 3^r$.

1. Recognise the sum as a binomial expansion: $(1 + 3)^{10} = \sum_{r=0}^{10}\binom{10}{r}\,1^{10-r}\,3^r = \sum_{r=0}^{10}\binom{10}{r}\,3^r$. 2. So the sum equals $(1 + 3)^{10} = 4^{10}$. 3. Compute: $4^{10} = 1{,}048{,}576$. 4. General pattern: $\sum_{r=0}^{n}\binom{n}{r}\,a^r = (1+a)^n$. 5. Option B is the sum with $a = 1$ instead of 3. Option C is one term times $n$. Option D is the middle term only. _Source: NCERT Class 11 Mathematics, Ch 7, §7.2.2 (Binomial coefficient sums), p. 5–6._