How many different five-digit numbers
can be formed using three 2s and two 5s?

Question

How many different five-digit numbers
can be formed using three 2s and two 5s?

amistre64 · Answer

theres a counting rule for this:$$\frac{(a+b+c+...+k)!}{a!b!c!...k!}$$

anonymous · Answer

never heard of this rule, can you elaborate please?

amistre64 · Answer

that was the elaboration

amistre64 · Answer

we have 2+3 elements that we are trying to order

we have 2 of one element, and 3 of the other

amistre64 · Answer

how many ways are there to arrange:  aaabbbcc?

8 elements, 3 are the same and 3 are the same and 2 are the same

$$\frac{(3+3+2)!}{3!3!2!}$$

anonymous · Answer

oh okay i get it, one sec let me try it with question

anonymous · Answer

Got it thanks :D

amistre64 · Answer

youre welcome, and good luck ;)

anonymous · Answer

oh btw what is this rule called?

anonymous · Answer

oh wait nvm i found the rule in my book, in the book is actually $$n!/a!b!c!...$$

amistre64 · Answer

hmm, ive always known it as the counting rule .... http://www.hamilton.ie/ollie/EE304/Counting.pdf around slide 11

amistre64 · Answer

yeah, its in a book too :)

amistre64 · Answer

the n part is the number of elements, which is just the sumof its parts

anonymous · Answer

Right, anyways thanks for the help :)

amistre64 · Answer

yw

amistre64 · Answer

its a more generalized version of the permuation rule that we start with.

take the the example:  abcde  how many ways can we arrange these?  5! ways of course but it fits in the formula as:$$\frac{(1+1+1+1+1)!}{1!1!1!1!1!}=\frac{5!}{1}$$