scenic tiger
Hi there! Can someone please help with this question? I don’t know if I’m just overthinking but it’s really confusing me!
How many 3-digit combinations can be formed out of digits 1, 2, 3, 4, 5 if at least one digit MUST be repeating?
I know that if you want to make a 3-digit combo out of the above numbers and they CAN repeat, the answer is 5 x 5 x 5 = 125. There are 125 possible combinations, but how do I calculate it if AT LEAST ONE MUST repeat?
Thanks so much in advance!
tulip cairnBOT
charred osprey
Since each combination will have only 3 digits, only one digit can repeat per combination
so you can have 555 or 551 or 511
In each of these cases, only one digit can repeat
Not more than one
so for each combination you have 5 different digits that can be chosen for repetition
As you have 3 spaces for placing the digits, you have 5 choices for repetition in each space
so youd multiply 5×5×5 since the cases arent exclusive
Thats about as good as i can explain
vital dock
i'd approach it as generating a 2 digit number from 5 cause only 2 digits need to be picked and then using the counting principle to multiply it out
scenic tiger
As you have 3 spaces for placing the digits, you have 5 choices for repetition i...
So I would treat the question as if it were just saying that digits can repeat therefore 5 x 5 x 5? Because in this cause the combo could be 123, 134, 425, etc. if one of the numbers has to be the same, would it not be 5 x 1 x 5 giving us 25 possible combos that have at least one digit that has repeated?
charred osprey
So I would treat the question as if it were just saying that digits can repeat t...
Actually, since at least one digit must be repeated, the combo cannot be 123 or 425. There must be 2 digits or 3 digits in the combo that are the same
scenic tiger
Actually, since at least one digit must be repeated, the combo cannot be 123 or ...
Yes so I’m just confused how I calculate that so that at least one digit repeats?
charred osprey
Yes so I’m just confused how I calculate that so that at least one digit repeats...
Alr lets exclude the cases where one digit is used three times
Theres 5 ways to do that
for the rest, you can write any digit twice with the remaining digit being something else
Such as 11_
here theres four ways
2,3,4 and 5 can go here
vital dock
i'd approach it as generating a 2 digit number from 5 cause only 2 digits need t...
would this work?
charred osprey
Instead of the digit 1, you could choose 2,3,4 or 5
that gives a total of five ways
finally you can still arrange the digits in a different order, given by 3!
so thats 5×4×3!
5 ways
From the cases
111, 222, 333, 444, 555
but instead of complicating, you can just write it as (number of given elements)^ordered arrangements
5³
vital dock
wait i think i found a simpler way:
number of total 3 digfit combinations formed from the 5 numbers - number of combinations where there is no repitition
using this i get 65
vital dock
how would you find the total number of combinations in the first place
isnt it just 125
because in each slot you have a choice between 5 numbers as long as we dont have restrictions, so 5^3 or 125
charred osprey
that would be the number of ways with repetition
charred osprey
total should just be 185
vital dock
no, its 125 i think
cause for each of the digits you can choose freely between any of the 5 numbers, using the fundamental counting principle its 125
charred osprey
hmm
scenic tiger
It would show the number of ways if repetition was allowed but not required. That would be 125. 5 x 5 x 5. But if a digit has to repeat so atleast two numbers have to be the same, I’m not sure how to calculate that
vital dock
It would show the number of ways if repetition was allowed but not required. Tha...
i used the principle of inclusion-exclusion
its easy to calculate number of digits with no repitition at all
scenic tiger
Yeah that’s easier cause it’s 5 x 4 x 3
scenic tiger
Okay so 65?
vital dock
yeah
charred osprey
I doubt it
Because
The question said that at least one digit must repeat
But theres only 3 available spaces
No way 2 digits can repeat
scenic tiger
I’m about to write out every possible combo 😂
vital dock
but i dont see why my reasoning doesn't stand.
total number of combinations = number of combinations where repititon is allowed + where repition isnt
charred osprey
All they asking for is that it must repeat
charred osprey
but i dont see why my reasoning doesn't stand.
total number of combinations = n...
For this question, they indirectly said that no repetition is not allowed
At least one digit must repeat
scenic tiger
Yes
charred osprey
So it cant 123
vital dock
yeah it cant
charred osprey
All 3 digits cant be different
vital dock
yeah
charred osprey
?
vital dock
is whats in the question
vital dock
oh nvm i thought u meant a digit can repeat once
vital dock
thats the total number of digits, with and without repitition
vital dock
No way 2 digits can repeat
i dont see how this is relevant to my solution
charred osprey
thats the total number of digits, with and without repitition
Maybe because this aint right
the part where you cubed 5 because you had 5 options for each space doesnt account for permutations without repetition
vital dock
the part where you cubed 5 because you had 5 options for each space doesnt accou...
yeah because im counting total permutations
with and without repititon
charred osprey
Write down every permutation that has at least one digit repeating
I guarantee itll be 125
vital dock
no thats too much work
charred osprey
Not for @scenic tiger
vital dock
well
can you tell me why the total number of digits isn't 125?
because using the fundamental principle of arithmetic, it very much is
brain fart moment wait
i meant counting
charred osprey
Instead of the digit 1, you could choose 2,3,4 or 5
This
I calculated the value 125 accounting for repetition only
vital dock
I calculated the value 125 accounting for repetition only
can you show me your entire calculation
or just type it out
charred osprey
like
scenic tiger
Thanks guys, it’s just confusing because of the wording. If one digit haaas to repeat it isn’t the same as if a 3 digit number just /could/ have a repeat
vital dock
suppose i take two spaces to be filled out by one repeating digit
no no
im not asking you that
tell me the total number of permutations
charred osprey
185
vital dock
how so
charred osprey
alr we can agree that without repetition theres 60 ways right?
vital dock
yeah
charred osprey
i comprehensively typed out the process i used to get 125 ways accounting for one digit being repeated
above
it doesnt account for the number of permutations without repetition
Theyre mutually exclusive
vital dock
As you have 3 spaces for placing the digits, you have 5 choices for repetition i...
i think this statement is flawed
because the choice in each space will be changed by your previous choices
for example 31_
you have 2 choices
scenic tiger
1 or 3 right?
charred osprey
Yes
vital dock
1 or 3 right?
yup
charred osprey
That was a shortcut way of thinking to avoid this rigmarole
you could permute other values for the first two spaces and have two choices for each set
and youd still end up with 125 permutations
vital dock
and youd still end up with 125 permutations
total
not repetitions
i think the problem with your solution is that
113 where the middle one is picked last
113 where 3 is picked last
is considered different outcomes
charred osprey
Alr lets exclude the cases where one digit is used three times
this method provides the same result of 125 ways
But with repetition only
vital dock
for the rest, you can write any digit twice with the remaining digit being somet...
1 can go here too
charred osprey
I excluded those 5 cases
where all three digits are the same
i worked out the ways where only two digits are the same
vital dock
Instead of the digit 1, you could choose 2,3,4 or 5
i lost you after this
charred osprey
I could repeat the same process using 22_ and 33_
vital dock
ok wait lets have a top down approach
total number of 3 digit numbers which can be formed from 5 digits is?
charred osprey
well our answers clearly differ
scenic tiger
Kay I just wrote it out
scenic tiger
25 total
charred osprey
You arent accounting for stuff like 414 313
Gonna be a long day for you
Or night
scenic tiger
lol
vital dock
111-115
121-125
131-135
141-145
151-155
these are all the numbers possibile with the first digit being 1
agreed?
scenic tiger
Okay so 313 isn’t the same as 331 in this question then I guess
charred osprey
Nope
vital dock
give me a counterexample.
charred osprey
these are all the numbers possibile with the first digit being 1
Alr
charred osprey
Yes
scenic tiger
Yep
vital dock
et voila
vital dock
Okay I’m just saying it’s 125
im talking about repititon + non repititon allowed
hold on emilyy
scenic tiger
Okay okay
charred osprey
and none of these permutations included a single 3-digit combination without repetition
vital dock
that's the total number of possible permutations ignoring repitition
vital dock
and none of these permutations included a single 3-digit combination without rep...
bro omg
they're included in the ranges
for example
vital dock
perhaps
vital dock
It just seems like a trick question with the way it was worded
whe emily did you understand all parts of my solution?
or any
scenic tiger
Yes I understand your reasoning for taking all possible permutations and then subtracting those that have no repeats
That makes the most sense to me giving us 125-60
vital dock
Yes I understand your reasoning for taking all possible permutations and then su...
did you understand how i got those individual values?
charred osprey
lets wait for @dire comet 's opinion
scenic tiger
😂
vital dock
dire comet
A) assuming repetition means the same digit is used twice in the combination
|at least one repetition| = |total| - |no repetitions|
= 5^3 - (5)(4)(3)=125-60=65
B) assuming repetition means the same digit used twice in a row
we assume two cases
i) 2nd repeats first (5)(1)(5)
ii) 2nd is different than first (5)(4)(1)
25 + 20 = 45
vital dock
@dire comet
scenic tiger
See that’s the part that’s hard. Does the repeat have to be one after the other or just 2 of the same digits in the 3 digit combo. I think 65 is the answer they are looking for
vital dock
See that’s the part that’s hard. Does the repeat have to be one after the other ...
same
charred osprey
time to ragequit at 4 am
vital dock
time to ragequit at 4 am
bro u almost made me break my keyboard
charred osprey
Truth is
dire comet
it's not clear, the question is flawed.
as a mathematician, be a well commented function and state all your assumptions
vital dock
Hi there! Can someone please help with this question? I don’t know if I’m just o...
i think they're looking for 65 because arrangement of repeated degits isnt specified
digits*
dire comet
if you're going to guess, state your assumptions, then you can't be wrong
scenic tiger
Okay perfect thanks everyone so much!!
dire comet
❤️
scenic tiger
❤️❤️❤️
unkempt dock
.solved