83. Let ⁿC_r denote the binomial coefficient

ⁿC_r = n! / [r!(n-r)!].

For n = 100, find the sum of the series

1 − ⁿC₁ + ⁿC₂ − ⁿC₃ + ··· + (−1)ⁿⁿC_n.

(A) 0

(B) 1

(C) 2

(D) 1024

Sum of the Binomial Coefficient Series 1 − nC₁ + nC₂ − nC₃ + … + (−1)ⁿ nC_n for n = 100

Understanding the Problem

The alternating positive and negative signs indicate that we should think about the expansion of (1 − 1)ⁿ. Once this identity is recognized, the entire problem becomes extremely simple.

Step 1: Recall the Binomial Theorem

The Binomial Theorem states that

(a + b)ⁿ = Σ ⁿC_r a^n-rb^r

where r = 0, 1, 2, …, n.

If we substitute

a = 1

and

b = -1,

then the expansion becomes

(1 − 1)ⁿ = ⁿC₀ − ⁿC₁ + ⁿC₂ − ⁿC₃ + ··· + (−1)ⁿⁿC_n.

Step 2: Compare with the Given Series

The given expression is

1 − ⁿC₁ + ⁿC₂ − ⁿC₃ + ··· + (−1)ⁿⁿC_n.

Since

ⁿC₀ = 1,

the given series is exactly the expansion of

(1 − 1)ⁿ.

Step 3: Substitute n = 100

For n = 100,

(1 − 1)¹⁰⁰ = 0¹⁰⁰.

Since

0¹⁰⁰ = 0,

the required sum is

0.

Final Answer

Option (A) = 0

Why This Method Works

The alternating pattern of the series is the key observation. Whenever a series contains binomial coefficients with alternating positive and negative signs, it should immediately remind us of the expansion of (1 − 1)ⁿ. Instead of computing hundreds of terms separately, the Binomial Theorem allows us to evaluate the complete expression in a single step. This identity is one of the most frequently used shortcuts in algebra and combinatorics.

Key Concepts Covered

Binomial Coefficient

The binomial coefficient is defined as

ⁿC_r = n! / [r!(n-r)!].

It represents the number of ways of selecting r objects from n distinct objects.

Binomial Theorem

The Binomial Theorem expands powers of a binomial as

(a+b)ⁿ = Σ ⁿC_ra^n-rb^r.

By choosing appropriate values of a and b, many complicated summations can be evaluated instantly.

Special Identity

A very important identity obtained from the Binomial Theorem is

Σ (−1)^rnC_r = (1−1)ⁿ = 0,

which holds for every positive integer n.

Conclusion

This question is an excellent example of how recognizing a standard algebraic identity can save significant time during competitive examinations. Instead of expanding or calculating individual binomial coefficients, identifying the expression as the expansion of (1−1)¹⁰⁰ immediately gives the answer.

Correct Answer: Option (A) = 0