What is the probability of the following event: given a five digit number x, the the sum of the digits of x is equal to 9.
The given answer is: C(12, 8)/90000
How is this answer arrived to?
2026-04-04 13:43:57.1775310237
What is the probability of the following event: given a five digit number x, the the sum of the digits of x is equal to 9.
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Five digit numbers run from $10000$ to $99999$ so the denominator is the total count of $N = 99999-10000+1 = 90000$.
By definition, the leading digit has to be at least $1$, so the rest must sum to $8$. The question becomes:
Imagine the target total of $8$ being eight stars $\star\star\star\star\star\star\star\star~$, and we have $5$ slots (digits) created by $4$ bars like $\star|\star\star||\star\star\star\star|\star$
The bars being adjacent to each other represents the digit being $0$ (nothing in that slot). The example just above represents the $5$ slots being $\{1,2,0,4,1\}$, which corresponds to the $5$ digit number of $22041$ with the leading $1$ added back.
Thus, $8$ starts and $4$ bars lining up for a combination count of $\dfrac{(4+8)!}{4!~ 8!}$, which in your notation is $C(12, 8)$ or $C_8^{12}$. This is your numerator.