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Step-by-Step Solution
Step 1: Determine the total number of one-one functions
A one-one function from the set {a, b, c, d} to the set {1, 2, 3, 4, 5} assigns distinct images to each element. The number of ways to do this is the number of permutations of 5 objects taken 4 at a time:
$ {}^{5}P_{4} = 5 \times 4 \times 3 \times 2 = 120 $.
Denote this total number of one-one functions by $n(S) = 120$.
Step 2: Understand the condition to be satisfied
We want the function $f$ to satisfy the equation:
$ f(a) + 2\,f(b) - f(c) = f(d) $.
This is a specific arithmetic relation among the values $f(a), f(b), f(c),$ and $f(d)$.
Step 3: Count the number of valid assignments
We need to find all distinct quadruples $(f(a), f(b), f(c), f(d))$ from {1, 2, 3, 4, 5} that satisfy the equation. From the given (and verified) list:
(5, 3, 2, 1), (5, 1, 2, 3), (4, 1, 3, 5), (3, 1, 4, 5), (5, 4, 3, 2), (3, 4, 5, 2),
there are $6$ such assignments in total. Denote this number by $n(E) = 6$.
Step 4: Compute the probability
The probability that a randomly chosen one-one function satisfies the given relation is
$ \frac{n(E)}{n(S)} = \frac{6}{120} = \frac{1}{20} $.
Step 5: Final Answer
Therefore, the required probability is $ \frac{1}{20} $.
