Proof The following rational 60%number

i think you meant to say proof that 60% is a rarional number. So like yeah, 60% is basically 60/100. if you reduce it by diving both by 20 it will become 3/5 or 0.60. we see that the fraction does not have specific decimals. whereas something like 0.732863821 is just random so its irrational.

there is only one way to arrange all the eight relative to each other because they are not distinct. we have

now there are nine places to place an (and not more than one in one place).   because we have four symbols, we need to choose four of these nine places to insert an . order does not matter because the are not distinct.

thus there are 1 · c(9,4) ways, or, if we calculate it,

thus there are 126 ways to form such "words."