How many permutations of the digits 1; 2; 3; 4; 5, have at least one digit in its
own spot? In other words, a 1 in the rst spot, or a 2 in the second, etc. For
example, 35241 is OK since it has a 4 in the fourth spot, and 14235 is OK, since
it has a 1 in the rst spot (and also a 5 in the fth spot). But 31452 in no good.
Hint: Let A1 be the set of permutations that have 1 in the rst spot, let A2 be the
set of permutations that 2 in the second spot, and so on.

Question

How many permutations of the digits 1; 2; 3; 4; 5, have at least one digit in its
own spot? In other words, a 1 in the rst spot, or a 2 in the second, etc. For
example, 35241 is OK since it has a 4 in the fourth spot, and 14235 is OK, since
it has a 1 in the rst spot (and also a 5 in the fth spot). But 31452 in no good.
Hint: Let A1 be the set of permutations that have 1 in the rst spot, let A2 be the
set of permutations that 2 in the second spot, and so on.

anonymous · Answer

fixing a single number in their own position, the number of permutations are obtained to be 4!.
fixing two numbers in their own position, the number of permutations are obtained to be 3!.
fixing three number in their own position, the number of permutations are obtained to be 2!.
fixing four numbers in their own position, the number of permutations are obtained to be 1!. which is the same as fixing 5 numbers in their respective positions.
so the total value comes out to be: 5*4!-10*3!+10*2!-5*1+1=66