Nine cards are numbered 1, 2, 2, 3, 3, 4, 6, 6, 6.Three of the nine cards are chosen and placed in a line, making a 3-digit number. Find how

Question

many different numbers can be made in this way if the number is between 200 and 300.

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asked by Raj
today at 12:27pm
There are three digits in your number ___ ___ ___
We sometimes refer to them as "slots"
The first slot is where the hundreds is placed.
For your number to be between 200 and 300 only the 2's could go in that slot
_ 2_ , ___, ___
Then there are 8 other numbers that could go in the next slot (since one of the two 2's is used up in the first slot), then 7 numbers left for the next slot
2 x8 x 7

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👩‍🏫
Ms Pi 3.14159265358979323
today at 12:53pm
be sure to adjust for duplicate digits, since the equal digits are indistinguishable.

The answer should be 22.

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asked by Raj
today at 2:14pm
Brute force gets you the answer of 22 : )
I am working on a formula method right now.
But here is brute force : )
212, 213, 214, 216, 221, 223, 224, 226, 232, 233, 231, 234, 236, 241, 242, 243, 246, 261, 262, 263, 266, 264

Can you tell me formula method?

Answer 1

It really doesn't matter that there are multiples of 6, 3, etc until we get to the cases
the only numbers possible for picking are 1,2,3,4,6
To have a 3 digit number that lies between 200 and 300 implies that the
number must take the form 2qq
There are doubles for 2,3, and 6
case 1: 2qq, where qq is one of the doubles
2 such cases , namely 233, 266
case 2: 2qq, where the qq is made up of 2 different ones
(1)(5 possible #'s)(4 possible #'s) <---- can't have two the same, that was case1
= 1()(5)(4) = 20

total number of cases = case1+ case2 = 2+20 = 22

there is no formula that will get you the 22 directly.