How Can I Improve My Skills in Permutations and Combinations?

[schumi1991`]

Although I have completed my high school and I will be going to college, there is one thing that i really regret about my high school and that is my weakness in solving permutation and combination problems i never had a proper guidance and whatever i could gather from books I tried and yet would fall short of finding the proper answer about 90% of the time.
So Please help me and tell me the right methodology in solving and the right approach... i would be grateful to you!

 

[mrbohn1]

I'm afraid you'll have to be a bit more specific in your request! I would say though, that practice makes perfect. It's a cliche, but you need to do a lot of problems at the same time as learning theory. Someone once said that "mathematics is not a spectator sport", or something to that effect.

Two books I would recommend on the subjects you mention are Concrete Mathematics by Graham, Knuth and Patashnik; and Combinatorics: Topics, Techniques, Algorithms by Peter J. Cameron.

In a broader sense, you might consider reading George Pólya's classic: How to Solve It.

Don't worry! I have worried about my problem-solving capabilities in the past, as I'm sure have many mathematicians. But everyone learns with time and practice.

 

[awkward]

The books suggested by mrbohn1 are excellent, but you might also take a look at one of my favorites, "The Mathematics of Choice, Or How to Count Without Counting" by Niven, which is at a more elementary level.

 

[Mentallic]

While I agree with mrbohn1's claim that practice makes perfect, I too have found practising combinatorics doesn't quite sink in as well as other topics in mathematics do for me. You must be in the same seat as me here (I still can't answer even the simple questions without hesitation).

 

[alice22]

I think it might help if you go back to basics and work from there.

Sometimes it is better to try to work out a few for yourself, without using formulas.

I learned much of what I know from working out combination and perms by hand gambling on horses, which helped quite a bit but here were some gaps.

For example the number of double from ten horses.

I start off

A horse could win any race ie

x000000000
0x00000000
00x0000000
and so on to
00000000x0
000000000x

That's 10 possibilities, now the second horse could win any of the remaining 9 races
ie for the first example, x000000000, it could be
xx0000000
x0x000000
x00x00000
to
x0000000x

So we have 10x9.

However there is a problem as some of those will be duplicates ie you get xx0000000 when the first horse wins the first race, but you also get xx0000000 when the first horse wins the second race, ie from doing all the possibilities of the second horse when the first horse wins race 2, ie 0x0000000, which also will give a line of xx0000000.Perhaps a simpler way of seeing this is to go to a really simple example, of finding the number of doubles with two horses. the answer is obviously one but my method would initially give starting points of

x0 and
0x

The first one has one double xx, as does the second also xx, however it is obvious they are the same, it might not be so easy to see that is a larger number of races because it is not as immediately obvious.
So I need to take this into account, I know there are two possibilities so I need to divide by 2.

If I was doing trebles it would be 10x9x8, then I need to eliminate the dupulicates from 3 horses, so how many are there? We can have 123, 132, 231, 213, 312, 321, that's 6 in total
so I divide by 6, which is 3! (factorial).

So it is easy to get the formula, ie n!/(r!(n-r)!

Knowing how to do that should be useful in other problems.

Actually I just found this, which basically explains the same thing.

http://betterexplained.com/articles/easy-permutations-and-combinations/

A better explanation than mine I think!

 

[Redbelly98]

Although I have completed my high school and I will be going to college, there is one thing that i really regret about my high school and that is my weakness in solving permutation and combination problems i never had a proper guidance and whatever i could gather from books I tried and yet would fall short of finding the proper answer about 90% of the time.
So Please help me and tell me the right methodology in solving and the right approach... i would be grateful to you!

What are your current plans, if any, for a major in college? If you go into the sciences or engineering, doing permutations and combinations really will not come up very often at all. Maybe that can help relieve some of the pressure or anxiety.