Prove that: `P(1,1)+2. P(2,2)+3. P(3,3)++ndotP(n , n)=P(n+1,\ n+1)-1.`

Explanation

To prove the equation \( P(1,1) + 2 \cdot P(2,2) + 3 \cdot P(3,3) + \ldots + n \cdot P(n,n) = P(n+1, n+1) - 1 \), we will analyze both sides step by step.

### Step 1: Understanding the Left-Hand Side (LHS)
The left-hand side consists of terms of the form \( k \cdot P(k,k) \) for \( k \) from 1 to \( n \).

Recall that \( P(k,k) = k! \) (the number of ways to arrange \( k \) items).

Thus, we can rewrite the LHS as:
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