Number of functions f:{1,2,…,100}→{0,1}, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:
Get the complete, step-by-step math solution for: "Number of functions f : \{1, 2, 100\} \{0, 1\}, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Understand the function definition
The function f maps elements from the set A={1,2,…,100} to the set B={0,1}. This means for each element in the domain, the function can assign either 0 or 1.
Step 2: Identify the constraint
The problem states that exactly one of the positive integers less than or equal to 98 is assigned the value 1. This means we need to choose one element from the set {1,2,…,98} to be mapped to 1.
Step 3: Determine choices for elements up to 98
Since exactly one integer from {1,2,…,98} must be assigned 1, there are 98 possible choices for which integer gets the value 1. All other 98−1=97 integers in this range must be assigned 0.
Step 4: Determine choices for remaining elements
The constraint only applies to integers less than or equal to 98. For the remaining integers, 99 and 100, there is no restriction. Each of these two integers can be assigned either 0 or 1 independently.
Step 5: Calculate total number of functions
We multiply the number of choices for each part of the domain. There are 98 ways to choose the integer less than or equal to 98 that gets 1. There are 2 choices for f(99) (0 or 1) and 2 choices for f(100) (0 or 1).