Consider the 10-digit number M = 9876543210. We obtain new 10-digit number from M according to the following rule: we can choose one or more disjoint pairs of adjacent digits in M and interchange the digits in these chosen pairs, kee the remaining digits in their own places. For example, from M = 9876543210, by interchanging the 2 underlined pairs, and kee the others in their places, we get M1 9786453210 . Note that any number of (disjoint) pairs can be interchanged. Find the number of new numbers that can be so obtained from M.
Hint 1: Start by analyzing the initial conditions and setting up the basic equations. Number of ways if single pair is changed = 9 = 9C1.
Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. Number of ways if 2 pairs changed = 7 + 6 + 5 + … + 1.
Hint 3: Proceed with the final algebraic steps to solve the system. Number of ways if 3 pairs changed solve for the final valueC3.
Step 1: Number of ways if single pair is changed = 9 = 9C1
Step 2: Number of ways if 2 pairs changed = 7 + 6 + 5 + … + 1
Step 3: Number of ways if 3 pairs changed = 7C3
Step 4: So, total numbers that can be formed = 9C1 + 8C2 + 7C3 + 6C4 + 5C5
Step 5: = 9 + 28 + 35 + 15 + 1
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