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$$\large B(t) = \sum_{i=0}^{n} \binom{n}{i}(1-t)^{n-i}t^{i}P_{i} $$
$$\large \text {where, parameter t:}$$
$$\large 0\leq t\leq 1 $$
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$$\large \binom{n}{k} = \frac{n!}{k!(n-k)!} = \prod_{i=1}^{k}\frac{n+1-i}{i} $$
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$$\large B(t) = (1-t)^3P_{0} + 3(1-t)^2tP_{1} + 3(1-t)^2t^2P_{2} + t^3P_{3} $$
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$$\large B(t) = (1-t)^5P_{0} + 5t(1-t)^4P_{1} + 10t^2(1-t)^3P_{2} + 10t^3(1-t)^2P_{3} + 5t^4(1-t)^4P_{4} + t^5P_{5} $$
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