1. How many even numbers can be formed if the digits of the number 112433 are rearranged?
2. In arranging a 10-day examination time-table involving 10 subjects and one subject per day, a teacher plans to have Mathematics, Physics and Chemistry all separate by at least one day. How many ways are possible?
Anyone please explain these to me, thank you.
PS: The answer for question 1 is 120 and the answer for question 2 is 1693440.
The given answer is incorrect.
Originally Posted by alexander9408
To be even the string must end in 2 or 4.
The string can be arranged in ways.
Add a 2 to the right-hand end of each of those 30 strings.
That is is an even number.
Now swap the 4 and the 2 from each of above that gives another 30.
So 60 in all.
BTW: there are 120 odd numbers.
If you have n distinct object, they can be arranged in n! different ways. If m of those things are the same, divide by m! to account for the fact that for each way of arranging the n things, m! of them are identical except for rearrangements of just those things. For example, there are ways to arrange the letters aabcdd. Apply that to your problem. (The correct answer is NOT 120.)
There are 3 ways in which we could have Mathematics, Physics, and Chemistry on the first day and 7 ways in which not.
If any one of Mathematics, Physics, or Chemistry were given on the first day, then there must be one other 7 courses. So far the number of different ways is 3(7)= 21.
If it was one of the other courses on the first day, there are 6 ways we can give a "non math, physics, chem" test- 7(6)= 42- and 3 ways we can give one of those courses:7(3)= 21.
If it was one of math, physic, chem on the second day, it must be one of the other 6 courses: 7(3)(6)= 126
Continue like that. Add the separate possible combinations to find the total.