How many hexadecimal chains of numbers of length 6 are there with an "A" in the first position or with "F"...?

Question

How many hexadecimal chains of numbers of length 6 are there with an "A" in the first position or with "F" in the 3rd and 4th position or a "0" in the last position? Only chains of numbers with at least one hex value less than 9 should be counted. The or link is not exclusive.

My guess:

First I use "Inclusion-exclusion principle" for the ""A" in the first position or with "F" in the 3rd and 4th position or a "0" in the last position" $\rightarrow(A \cup B \cup C ) = |A| +|B|+|C|-|A \cap B| -|A \cap C|-|B \cap C| + |A \cap B \cap C|$

|A| = $16^{5}$

|B| = $16^{4}$

|C| = $16^{5}$

solution (formula): = $2089216$

Now I don't know how I can get the result for the "hex value less than 9".

I wrote a short program that gave me the solution of $2070352$, but as I said I don't know how I can get the right solution.

#sendhelp

You also have to add $|A \cap B \cap C|$ in the formula for $|A \cup B \cup C|$. — Vishu, Apr 24 '20 at 20:33

score 1 · Answer 1 · answered Apr 24 '20 at 20:35

1

Hint: if $D$ is the set of chains with at least one hex value less than $9$, then $D$ is the complement $D'$ of the set of chains where all values are at least $9$. $D'$ has $7^6$ elements, so $D$ has $16^6 - 7^6$ elements.

answered Apr 24 '20 at 20:35

Rob Arthan

48,577

I've came to this conclusion too. But how do I calculate $D \cap (A \cap B \cap C)$? – drey Apr 24 '20 at 21:13
By more applications of the inclusion-exclusion principle. – Rob Arthan Apr 24 '20 at 21:49
how does $A \cap D$ look like? Cause I also tried that, but I got $16^{5} - 7^{6}$ and this is wrong I believe. – drey Apr 24 '20 at 21:58

How many hexadecimal chains of numbers of length 6 are there with an "A" in the first position or with "F"...?

1 Answers1