Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
Number of functions defined from $f:{\{1,2,3,4,5,6\} \to \{7,8,9,10\}}$ such that the sum $f(1)+f(2)+f(3)+f(4)+f(5)+f(6)$ is odd, is:
Step-by-Step Solution
Key Concept: By symmetry between odd and even outputs (2 odd, 2 even in codomain), exactly half of all possible functions yield an odd sum.
For the sum $f(1)+f(2)+f(3)+f(4)+f(5)+f(6)$ to be odd, we need an odd number of odd values among the function outputs. The codomain $\{7,8,9,10\}$ contains 2 odd numbers (7,9) and 2 even numbers (8,10). For each of the 6 domain elements, we can choose from 4 values in the codomain. The sum is odd when an odd count (1, 3, or 5) of the six function values are odd. By symmetry, exactly half of all $4^6 = 2^{12}$ functions produce an odd sum, giving $\frac{2^{12}}{2} = 2^{11}$ functions.
Correct Answer: 2