+0  
 
-1
61
8
avatar+48 

1.How many distinct subsets of the set S=1,8,9,39,52,91 have odd sums? (As an example, 1,8,52 is one such subset, because 1+8+52 equals 61, which is odd.)

 

2.How many distinct subsets of the set S=1,8,9,39,52,91 have three-digit sums? (As an example, 39,91 is one such subset, because 39+91 equals 130, which has three digits.)

 

3.Consider the set S=1,2,3,4,5,6,7,8,9,12,13,14,15,16,17,18,19,23,24,. . . . ,123456789\}, which consists of all positive integers whose digits strictly increase from left to right. This set is finite. What is the median of the set?

 

4.How many paths of minimum length are there from A to b in the grid below?

5.Consider the sequence 1,3,4,9,10,12,13,... which consists of every positive integer that can be expressed as a sum of distinct powers of 3. What is the 75th term of this sequence?

 

THANK YOU SO MUCH

lololololololololll  Jul 21, 2018
 #1
avatar+87564 
+1

4.How many paths of minimum length are there from A to b in the grid below?

 

 

Notice that if we consider a move downward to the right as "Southeast" = "SE"  and a move  upward to the right as "Northeast" = "NE," we have the following possible set  of moves from A to B :

 

( SE,SE, SE, NE, SE, NE, NE, NE )

 

So...the total number of  possible minimum paths is given by choosing any four of eight positions in the set for "SE"  (or, alternatively, choosing any four of eight positions in the set for "NE" ) = 

 

C(8, 4)  =   70  possible minimum paths

 

 

cool cool cool

CPhill  Jul 21, 2018
 #2
avatar
0

To get the solution to the problem, you need to subtract 2 from your answer- 70-2=68.

 

this is because the path (SE, SE, SE, SE, NE, NE, NE, NE) and the path (NE, NE, NE, NE, SE, SE, SE, SE) are not possible on the grid.

Guest Jul 21, 2018
 #5
avatar+87564 
0

Thanks, guest for spotting that !!

 

cool cool cool

CPhill  Jul 21, 2018
 #7
avatar
0

Thanks so much

Guest Jul 23, 2018
 #3
avatar+108 
0

I also need help on the first two..... Also, isnt 4 and 5 from AoPS homework?

Max0815  Jul 21, 2018
 #6
avatar
0

Yes lol you are in my class

Guest Jul 23, 2018
 #4
avatar+87564 
+1

5.Consider the sequence 1,3,4,9,10,12,13,... which consists of every positive integer that can be expressed as a sum of distinct powers of 3. What is the 75th term of this sequence?

 

Note the pattern that emerges :

3^0  = 1                                                                                                         1

3^1  = 3    3^0 + 3^1 = 4                                                                           1        1

3^2 = 9   3^2 + 3^0 = 10   3^2 + 3^3 = 12   3^2 + 3^1 + 3^0 = 13         1      2        1

Note that the  number of terms - 7-  is just the sum of the first 2 rows  of Pascal's Triangle  [ the first entry is "Row 0" ]

 

Note that the next terms are

3^3  = 27

3^3 + 3^0 = 28

3^3 + 3^1  = 30

3^3 + 3^1 + 3^0 = 31

3^3 + 3^2  = 36

3^3 + 3^2 + 3^0 = 37

3^3 + 3^2 + 3^1  = 39

3^3 + 3^2 + 3^1 + 3^0  = 40

 

There  are 8 terms  which is the sum  of the elements of the  3 row of Pascal's triangle =  1  3  3   1

And each row adds 2^n  more  terms   [ where n is the row number ]

So...after the 4th row we have 15 + 2^4  =  15 + 16  =  31 terms

After the 5th row we have  31 + 2^5  =  31 + 32  = 63 terms

 

The first term in the 6th row  will represent 64 term  ....this will be  = 3^6   = 729

Notice that the next few terms are

3^6 + 3^0  =  730

3^6 + 3^1  = 732

3^6 + 3^1 + 3^0  = 733  

3^6 + 3^2 = 738  ....

And note that the terms in each case appended after the first term just follow the pattern of the first 15 terms

So...the 75th term  will be the sum of this first term  plus the 11th term in the above  series [ 31]  

 

So....the 75th term is    729  + 31  =  760

 

 

cool cool cool

CPhill  Jul 21, 2018
 #8
avatar
0

THANK YOU SO MUCH 

Guest Jul 23, 2018

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