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Evaluate the following: \ sum(r=1)^5\ ^5...

Evaluate the following: `\ sum_(r=1)^5\ ^5C_r`

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`sum_{r=1}^{5}\ ^{5} C_{r}=\ ^{5} C_{1}+\ ^{5} C_{2}+\ ^{5} C_{3}+\ ^{5} C_{4}+\ ^{5} C_{5}`
`Rightarrow sum_{r=1}^{5}\ ^{5} C_{r}=\ ^{5} C_{1}+\ ^{5} C_{3}+\ ^{5} C_{3}+\ ^{5} C_{1}+\ ^{5} C_{0}`
`[because\ ^{n} C_{r}=\ ^{n} C_{n-r}] `
`Rightarrow sum_{r=1}^{5} 5_{C_{r}}=2 times(frac{5}{1} times 4_{C_{0}})+2 times(frac{5}{3} times frac{4}{2} times frac{3}{1} times 2_{C_{0}})+5_{C_{0}}`
`[because\ ^{n} C_{r}=frac{n}{r}\ ^{n-1} C_{r-1}] `
`Rightarrow sum_{r=1}^{5} 5 C_{r}=10+20+1=31 . quad[because\ ^{n} C_{0}=1]`
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