Calculate the value of a complex combination expression involving multiple combination terms | Step-by-Step Solution

TinyProf's Step-by-Step Solution

1. What We're Solving: We need to calculate: 404C4 - 4C1 × 303C4 + 4C2 × 202C4 - 4C3 × 101C4

This looks complex, but there's a pattern that will make our work much easier!

2. The Approach: This expression has a very specific structure that suggests we should use the Binomial Theorem in reverse! Notice how:

• The first numbers decrease by 101 each time: 404, 303, 202, 101
• We're multiplying by binomial coefficients 4C0 (which is 1), 4C1, 4C2, 4C3
• The signs alternate: +, -, +, -

This pattern screams "binomial expansion"!

3. Step-by-Step Solution:

Step 1: Recognize the pattern Rewrite to make the pattern clearer:

• 4C0 × 404C4 - 4C1 × 303C4 + 4C2 × 202C4 - 4C3 × 101C4

Step 2: Look for the binomial connection

• 404 = 101 × 4 + 0
• 303 = 101 × 3 + 0
• 202 = 101 × 2 + 0
• 101 = 101 × 1 + 0

Step 3: Apply Vandermonde's Identity (Chu-Vandermonde Identity) This looks like it could be related to the identity: ∑(k=0 to r) (-1)^k × rCk × (a-kr)Cn = something special

Step 4: Recognize this as (a-b)^n expansion If we think of this as coefficients of (x-1)^4 applied to combinations, we can write: This equals: ∑(k=0 to 4) 4Ck × (-1)^k × (404-101k)C4

Step 5: Use the hockey stick identity insight This is a special case! When we have this pattern, it often equals: (404-101×4)C4 = (404-404)C4 = 0C4 = 0

4. The Answer: The value is 0, which corresponds to option (C).

5. Memory Tip: When you see alternating binomial coefficients multiplied by combinations where the "n" values decrease by a constant amount, think about whether the pattern might telescope to zero! The key insight is that these types of expressions often represent "net change" calculations that cancel out completely.

Great job working through this challenging combinatorics problem! The pattern recognition is the hardest part - once you see it, the mathematics becomes much more manageable.