Sequences combinatory

[MathIsEverything]

I'm struggling with this problem can you guys help me?
Let S be the set of all sequences (a, b, c, d, e) with terms belonging to the set of numbers {0,1, … 9}.

How many sequences are in the set

1. Which are decreasing​

​

2. Whose product abcde is an even number​

​

3.Whose sum of the digits of the product abcde is a number divisible by 9​

Thanks for responding

 

[Harry_the_cat]

What have you tried?

You need to know if a, b, c, d, and e can be the same number? or are they distinct?

 

[MathIsEverything]

I think they may be the same. I tried 2 point and figured out that the product abcde is not even if and only if each of the factors a, b, c, d, e is an odd number, i.e. 5x5x5x5x5 (because there are 5 odd numbers and they can repeat) and all sequences are 10x10x10x10x10, so the answer in point 2 should be 10^5 - 5^5. I do not know if Im thinking right. I haven't figured out how to do point 1 and 3 yet, and I am very interested in these problems.

 

[pka]

I'm struggling with this problem can you guys help me?
Let S be the set of all sequences (a, b, c, d, e) with terms belonging to the set of numbers {0,1, … 9}.
How many sequences are in the set
1. Which are decreasing
2. Whose product abcde is an even number
3.Whose sum of the digits of the product abcde is a number divisible by 9

We need to read the exact wording of the question.
For example: is it decreasing or non-increasing? In a decreasing sequence all terms must be distinct. That is not the case for non-increasing sequences.
[imath](9,7,5,2,1)[/imath] is a decreasing sequence.
[imath](8,7,7,2,2)[/imath] is a non-increasing sequence.
Which is it according to your text-book?
Please post the exact wording of the question.

 

[Dr.Peterson]

I think they may be the same. I tried 2 point and figured out that the product abcde is not even if and only if each of the factors a, b, c, d, e is an odd number, i.e. 5x5x5x5x5 (because there are 5 odd numbers and they can repeat) and all sequences are 10x10x10x10x10, so the answer in point 2 should be 10^5 - 5^5. I do not know if Im thinking right. I haven't figured out how to do point 1 and 3 yet, and I am very interested in these problems.

That's a good method for (2).

How many sequences are in the set

1. Which are decreasing

For (1), notice that any subset of the digits can be arranged to be strictly decreasing. (I'm guessing that is what they mean.)

How many sequences are in the set

3.Whose sum of the digits of the product abcde is a number divisible by 9

For (3), first, have you heard of "casting out nines"? What does that tell you about the factors of the numbers? Then, what must be true if that is not true of the product? You'll probably need a couple cases.

 

[MathIsEverything]

We need to read the exact wording of the question.
For example: is it decreasing or non-increasing? In a decreasing sequence all terms must be distinct. That is not the case for non-increasing sequences.
[imath](9,7,5,2,1)[/imath] is a decreasing sequence.
[imath](8,7,7,2,2)[/imath] is a non-increasing sequence.
Which is it according to your text-book?
Please post the exact wording of the question.

The exact wording of the question is: Let S be the set of all sequences (a, b, c, d, e) with expressions belonging to the set of numbers {0.1,..., 9}. How many sequences are there in the set S a) decreasing? b) whose product abcde is an even number? c) in which the sum of the digits of the product abcde in decimal notation is divisible by 9?
There's nothing about distinct or anything else

 

[MathIsEverything]

That's a good method for (2).


For (1), notice that any subset of the digits can be arranged to be strictly decreasing. (I'm guessing that is what they mean.)


For (3), first, have you heard of "casting out nines"? What does that tell you about the factors of the numbers? Then, what must be true if that is not true of the product? You'll probably need a couple cases.

For(3) I haven't heard about casting out nines, so I'll read about that in my free time. But I think I'll need help with that because I'm just starting with combinatory, but I'm trying learn more from books and internet.
For (1) Are you sure that any substet of he digits can be arranged to be strictly decreasing? For example (9,7,7,1,1) cannot be, or I misunderstood you? Thank you for your suggestions and time

 

[Dr.Peterson]

For(3) I haven't heard about casting out nines, so I'll read about that in my free time. But I think I'll need help with that because I'm just starting with combinatory, but I'm trying learn more from books and internet.

I think they probably expect you to know, if not the full idea of casting out nines, then the more basic fact that you can test for divisibility by 9 by looking at the sum of the digits.

For (1) Are you sure that any subset of he digits can be arranged to be strictly decreasing? For example (9,7,7,1,1) cannot be, or I misunderstood you? Thank you for your suggestions and time

Ah, but that isn't a subset! A subset has to consist of distinct elements!

But that's a great question to have asked, because it shows you're thinking.

 

[MathIsEverything]

I think they probably expect you to know, if not the full idea of casting out nines, then the more basic fact that you can test for divisibility by 9 by looking at the sum of the digits.

I was thinking about that and i know that number is divisible by 9 if sum of the digits is also divisible by 9. So there will be 4 cases where abcde od divisible by 9
1) sequences which includes one or more 0 or 9
2) sequences without any 0 or 9 which include two or more 3
3) sequences with no 0,9, with one 3 and one or more 6
4) sequences with no 0,9,3 and two or more 6
But i dont know how to calculate the number of this sequences can you tell me how to calculate each of the case?

 

[Steven G]

I think they may be the same. I tried 2 point and figured out that the product abcde is not even if and only if each of the factors a, b, c, d, e is an odd number, i.e. 5x5x5x5x5 (because there are 5 odd numbers and they can repeat) and all sequences are 10x10x10x10x10, so the answer in point 2 should be 10^5 - 5^5. I do not know if Im thinking right. I haven't figured out how to do point 1 and 3 yet, and I am very interested in these problems.

10^5 - 5^5 = 2^5*5^5 - 5^5 = 5^5(2^5 - 1) = 5^5 (31)= 31*5^5. Why 31?
Final answers really should sometimes be simplified as you might gleam sometime from that representative of the answer.

 

[Dr.Peterson]

Let S be the set of all sequences (a, b, c, d, e) with terms belonging to the set of numbers {0,1, … 9}.

How many sequences are in the set

3.Whose sum of the digits of the product abcde is a number divisible by 9

I was thinking about that and i know that number is divisible by 9 if sum of the digits is also divisible by 9. So there will be 4 cases where abcde od divisible by 9
1) sequences which includes one or more 0 or 9
2) sequences without any 0 or 9 which include two or more 3
3) sequences with no 0,9, with one 3 and one or more 6
4) sequences with no 0,9,3 and two or more 6
But i dont know how to calculate the number of this sequences can you tell me how to calculate each of the case?

First, to make it explicit, you see that the question can be restated as

How many sequences are in S whose product a*b*c*d*e is a multiple of 9?​

And you've identified several cases, namely those with at least one 0 or 9; and those with no 0 or 9 but with two or more numbers that are 3 or 6 (which you break into four cases).

Start with the first, for practice. How many sequences of five digits contain at least one 0 or 9? Give that a try. Your method for (2) suggests you know what to do.

 

[MathIsEverything]

Start with the first, for practice. How many sequences of five digits contain at least one 0 or 9? Give that a try. Your method for (2) suggests you know what to do.

10^5-8^5=67232?

 

[MathIsEverything]

And in a) i think its just the number of subsets of {0,1,2,3,4,5,6,7,8,9} od 5 elements, so newton symbol 10 over 5 which equals to 252 and can you give me na idea how to calculate other 3 cases in c)

 

[Dr.Peterson]

10^5-8^5=67232?

Yes.

And in a) i think its just the number of subsets of {0,1,2,3,4,5,6,7,8,9} of 5 elements, so newton symbol 10 over 5 which equals to 252 and can you give me an idea how to calculate other 3 cases in c)

Also correct for (1), since each subset of 5 corresponds to one decreasing sequence.

As for the rest of (3), keep thinking the way you already have! As I said, you want "those with no 0 or 9 but with two or more numbers that are 3 or 6".

I first found the number of sequences containing no 0 or 9 and no 3 or 6, then the number containing exactly one 3 or 6, and used subtraction once again.