Q.A lens cannot be seen when immersed in a transparent medium. When is this possible?

Index Matching: Why the Formula Holds

Index matching is a powerful technique in combinatorics and probability — it's used to simplify sums over complicated index sets by cleverly re-indexing or pairing terms. The core idea is to match indices so that a double sum (or product) collapses into a simpler expression.

Let's build the reasoning step by step.

The Core Formula

The most common index matching identity is:

This looks trivial — it's just the distributive law. But the why matters for deeper applications.

Why It Holds: The Distributive Law in Action

Step 1: Expand the outer sum

The left side means: for each fixed $i$, sum over all $j$, then sum over $i$.

Step 2: Factor out $a_i$ from the inner sum

Since $a_i$ does not depend on $j$, it can be pulled out:

Step 3: The inner sum is constant with respect to $i$

Let $S_b = \sum_{j=1}^{n} b_j$. Then:

Step 4: Recognize the product

This is exactly:

Key insight: The double sum over all $n^2$ pairs $(i,j)$ is just the product of the two separate sums. This works because the terms factor as $a_i b_j$ — no cross-dependence between $i$ and $j$.

Why This Matters for Exam Problems

Index matching is used when you have double sums with constraints (like $i < j$ or $i \neq j$). The trick:

1. Start with the unconstrained double sum (all $i,j$)
2. Subtract the diagonal terms ($i=j$) or the off-diagonal terms
3. Use index matching to simplify

Example: Sum over $i < j$

We want $\sum_{1 \le i < j \le n} a_i b_j$.

Derivation:

The first and third terms are symmetric (just swap $i$ and $j$). So:

Thus:

Why this works: The unconstrained double sum counts each unordered pair $(i,j)$ twice (once as $(i,j)$ and once as $(j,i)$), except the diagonal which appears once. Index matching lets us express the constrained sum in terms of the product.

The Deeper "Why": Symmetry and Factorization

The real power of index matching comes from symmetry: …