Let R={a, b, c, d, e} and S={1,2,3,4}. Total number of onto functions f: R → S such that f(a) ≠ 1, is equal to ______________.

Solution

Maths sir · 4 steps

Fix f(a) first, then check which values b, c, d, e must still cover so the map stays onto.

1. Choose f(a). f(a) can be 2, 3 or 4, so there are 3 choices. Take f(a) = 2; the other two choices work the same way.
2. Cover the rest. Now b, c, d, e must hit 1, 3 and 4. They may also hit 2, but they do not have to. Split into two cases: the four elements hit all of {1, 2, 3, 4}, or they hit exactly {1, 3, 4}.

   Values taken by b, c, d, e	How to count	Number of ways
   all of {1, 2, 3, 4}	4 elements onto 4 values: 4!	24
   exactly {1, 3, 4}	pick the two that share a value, C(4, 2) = 6, then arrange 3 values: 3! = 6	36
   total for f(a) = 2	24 + 36	60

3. Check by symmetry. Onto maps from 5 elements to 4 number 4⁵ - 4 × 3⁵ + 6 × 2⁵ - 4 = 240. By symmetry, f(a) equals each value in a quarter of them. So f(a) = 1 in 60 of them, and 240 - 60 = 180.
4. So. There are 3 choices for f(a) and 60 ways for each, which gives 3 × 60 = 180.

The common slip: 240. Ignoring the condition f(a) ≠ 1 counts every onto function and gives 240, not 180.

The idea to keep. When one element has a restriction, fix its image first. Then count how the remaining elements cover whatever values are still uncovered, allowing repeats of the fixed value.

Answer from the NTA final answer key. Solution by PrepWiser.