Question

Answer this kindly NO SPAM
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Answer 1

Answer:

2ⁿ - 1

Step-by-step explanation:

Method 1  ---  Binomial expansion

( 1 + x )ⁿ = ⁿC₀ + ⁿC₁x + ⁿC₂x² +... + ⁿCₙxⁿ

Putting x = 1, this gives

2ⁿ = ⁿC₀ + ⁿC₁ + ⁿC₂ +... + ⁿCₙ

Since the first term on the right is ⁿC₀ = 1, substracting 1 from both sides gives

2ⁿ - 1 = ⁿC₁ + ⁿC₂ +... + ⁿCₙ

Method 2  ---  Counting

Consider a set A with n elements and count the number of subsets in two ways.

First, as each of the n elements is either included or excluded from a subset (that's 2 options), the number of subsets is equal to 2ⁿ.

Second, the number of subsets is equal to

( # subsets with 0 elements ) + ( # subsets with 1 element ) + ( # subsets with 2 elements ) + ... + ( # subsets with n elements )

= ( # ways to choose 0 elements from n ) + ( # ways to choose 1 element from n ) + ( # ways to choose 2 elements from n ) + ... + ( # ways to choose n elements from n )

= ⁿC₀ + ⁿC₁ + ⁿC₂ +... + ⁿCₙ

Thus   2ⁿ = ⁿC₀ + ⁿC₁ + ⁿC₂ +... + ⁿCₙ.

Now finish as in Method 1.

Since the first term on the right is ⁿC₀ = 1, substracting 1 from both sides gives

2ⁿ - 1 = ⁿC₁ + ⁿC₂ +... + ⁿCₙ