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#### 1 Please prove the following using induction

###### Math

1 Please prove the following using induction.

n choose 0 = n choose n = 1 for all n greater or equal to 0

n choose k = n - 1 choose k - 1 plus n - 1 choose k for all 0 < k < n; n greater than or equal to 0

2. Please prove using the Inclusion and Exclusion Principle.

3. Prove using Pascal's formula.

C(n + 2, r) = C(n,r) + 2C(n, r - 1) + C(n, r - 2) for all 2 less than or equal to r which is less than or equal to n

4. a) Expand (1 + x)^n

b) Differentiate both sides of the equation from part (a) with respect to x to obtain:
n(1 + x)^n-1 = C(n, 1) + 2C(n, 2)x + 3C(n, 3)x^2 + ... + nC(n, n)x^n-1

c) Prove that C(n, 1) + 2C(n, 2) + 3C(n, 3) + ... + nC(n,n) = n2^n-1

d) Prove that C(n, 1) - 2C(n,2) + 3C(n, 3) - 4C(n, 4) + ... + (-1)^n-1 ? nC(n, n) = 0

5. a) Prove that (2^n + 1) / (n + 1) = C(n,0) + (1/2)C(n,1) + (1/3)C(n,2) + ... +
(1/n+1)C(n,n)

b) Prove that (1/n + 1) = C(n,0) + (1/2)C(n,1) + (1/3)C(n,2) + ... + (-1)^n ? (1/n + 1)C(n,n)

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