Combinatorics problem 4-d
Problem 4:
This year, you are helping organize your college’s career fest. There are 11 companies which are participating, and you have just enough room fit all of them. How many ways can you arrange the various firms, assuming…:
d) … Deutsche Bank representatives cancel, so you can give the additional space to one of the other companies?
Solution given:
We have 10 firms, which need to fill out 11 spots. Then, if we start filling up the room in some specific order, then there are going to be 10 options for who gets the first position. Since any firm can be given the additional space provided by DB’s withdrawal, then there are once again 10 options for the second spot. Then, there would be 9 different options for the third and so on. This results in having 10×10×9×8…×1=10×10!=36,288,000 many options to arrange the firms.
My question:
Please clarify why this is different from the following:
Take a set S of n different elements where the first element repeats n1 times, the second element repeats n2, ….. the kth element repeats nk times ( n1 + n2 + …… nk = n). Choose these elements in a specific order. Each such choice is called a permutation with repetition of the n elements. Two permutations are different if the elements are in a different order. The number of these permutations is:
Example: How many different numbers composed of 10 digits can we do with 5 ones, 2 twos, 2 threes and 1 four?
>>>>> I thought this was how I should approach this problem, hence the answer should be 11!/2! or 19,958,400.
To make it clearer, suppose 9 companies pulled out instead of just 1, and now you decided to give 5 spots to one of the remaining companies and 6 spots to the other. Using this formula, the answer would be calculated as 11!/5!x6! or 462. Using your explanation however, it would go 2 x 2 x 2 x 2 x 2 x 1 x .. 1 or 32, which seems wrong. However, I don't know enough theory to be able to easily tell if your answer is right or wrong, and if it's wrong, why it's wrong, and if it's right, why the answer I thought of was wrong in this instance.
I thought about it a bit and came up with some possible explanations, but I thought I'd ask you instead of spending time thinking about speculations.
Thank you very much.