Question: Numbers with at Least One 7

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why its only from 0 to 9 and not including all numbers until 100
greenlight-admin's picture

The question concerns 3-digit integers, and integers less than 100 do not have 3 digits. For example, 9, 28 and 77 do not have 3 digits.

Thank you very much for this helpful explanation. But I wonder could this rule works for the question below as well? I have tried so but it did not work. I wonder why?

"Four women and three men must be seated in a row for a group photograph. If no two men can sit next to each other, in how many different ways can the seven people be seated?"
greenlight-admin's picture

I'm not sure how we'd apply the approach used in the above video solution to your question.

Can you show me you solution so I can see if I can see what went wrong?

By the way, here's my solution to the question you posed: http://www.beatthegmat.com/tricky-counting-four-women-and-three-men-must...

My doubt here is- when we use strategy one or strategy 2, while explaining questions in the previous lessons, you said: After filling the most restrictive stage when the other stages are equally restrictive, we place no of ways in first stage then reduce one and place the number of ways in second stage- I mean if first stage is accomplished in 5 ways, the next stage will be accomplished in 4 ways as one object is used in first stage we are left with only 4 objects now. My question is why have you not applied that logic here? If I solve that way I get for strategy 2 (step 1) : 9*9*8 ways = 648 ways
and step 2: 8*8*7 ways =448 ways 648-448=200 ways. Please explain. Also please let me know if my question is not making sense.
greenlight-admin's picture

Quote: "After filling the most restrictive stage when the other stages are equally restrictive, we place no of ways in first stage then reduce one and place the number of ways in second stage"

The number of ways to complete each stage depends on the question. We can't take a strategy we use to solve one question and apply it to all questions. That seems to be what you are doing here.

As for your calculations, please tell me what each value represents. It's hard to know what you're doing when you write 8*8*7 ways

Cheers,
Brent

Okay. What I meant by 8*8*7 ways is for the step 2: which is break restriction. Now for the 1st digit we have 8 ways (with no 0 and no 7), The second stage can have 9 ways (including all 9 digits) as there is no restriction. This is explained in the video. My doubt is when one digit is used in 1st place, we are left with only 8 digits, so I took 8 in 2nd place and 7 in 3rd place. Hence I got 8*8*7. Now why I did this because, in the previous module question of arranging letters J,K,L,M and N, it is said that after filling the 3rd letter which is most restrictive, we fill the next places which are equally restrictive and we fill it like 4 ways, 3 ways 2 ways and 1 way. We do this because one letter is already used for 3rd place, now we have 4 letters left, so 4 ways and likewise. My question is there is nowhere mentioned in question that the letters cannot repeat. Taking that into account, can't we fill the other equally restrictive stages with 5 ways? The same way we did for the digits? Or vice versa, why are we not taking this principle for the digits that once a digit is used for 1 stage, the next stage will have one way lesser because 1 digit is used in 1st stage. Please clarify.
greenlight-admin's picture

Quote: "Okay. What I meant by 8*8*7 ways is for the step 2: which is break restriction. Now for the 1st digit we have 8 ways (with no 0 and no 7), The second stage can have 9 ways (including all 9 digits) as there is no restriction. This is explained in the video. My doubt is when one digit is used in 1st place, we are left with only 8 digits, so I took 8 in 2nd place and 7 in 3rd place. "

The above question does not state that digits cannot be repeated.

So, in step 2, we have 9 options for the tens digit (0, 1, 2, 3, 4, 5, 6, 8, or 9)
And we have 9 options for the units digit (0, 1, 2, 3, 4, 5, 6, 8, or 9)

In the question you alluded to (arranging letters J,K,L,M and N), we cannot repeat letters. So, once we choose a letter for the first position, we cannot re-use that letter. We know we cannot repeat letters, because we're told to ARRANGE the 5 letters. That's what makes that question different from this one.

Does that help?

Cheers,
Brent

I see, so the catch for the letter question is the word ARRANGE. So while arranging letters we cannot repeat them Is this understanding correct?
greenlight-admin's picture

That's correct.
So, for example, if you're asked to arrange the letters A, B and C, then all of the possible outcomes will have 3 letters AND no letters will be repeated.

The only exception is when there are duplicates among the letters to be arranged.
For example, if you're asked to arrange the letters in the word NOON, then all of the possible outcomes will have 4 letters AND each arrangement must have 2 N's and 2 O's.

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