Bosco

The above possiblity presupposes the two members have different titles, such as president and vice president.   

In such case, for example, Janet & Frank would be different from Frank & Janet.  If there are no differentiations  

between the two members, however, the possibilities would be halved, and the total number would be 91 ways.   

_.   

Starting with the A moving one letter at a time vertically, horizontally, or diagonally, how many different paths spell ARCH?   

        A   

     RRR   

  CCCCC   

HHHHHHHH   

From the A, there are three ways to get to an R.  

From the R, there are three ways to get to a C.    

From the C, there are three ways to get to an H.   

3 x 3 x 3  =  27 paths   

There are 8 H's in the bottom row.  That 8th one is superfluous.   

_.   

Catherine rolls a standard $6$-sided die six times. If the product of her rolls is $6,$ then how many different sequences of rolls could there have been? (The order of the rolls matters.)    

1 1 1 1 1 6      1 1 1 1 2 3      1 1 1 1 3 2     

1 1 1 1 6 1      1 1 1 2 3 1      1 1 1 3 2 1     

1 1 1 6 1 1      1 1 2 3 1 1      1 1 3 2 1 1 

1 1 6 1 1 1      1 2 3 1 1 1      1 3 2 1 1 1     

1 6 1 1 1 1      2 3 1 1 1 1      3 2 1 1 1 1     

6 1 1 1 1 1    

Looks like 16 ways to me, if I haven't missed any.     

_.   

A baking club wants to form an executive committee. There are $15$ people in the baking club, including Mark. In how many ways can the baking club form an executive committee with $2$ people, not including Mark?   

Excluding Mark, there are 14 members left to choose from.   

There are 14 choices for the first member of the committee.   

After the first is chosen, 13 choices are left for the second.    

14 x 13 = 182 ways to form the committee.    

_.

The parabola y = ax^2 + bx + c is graphed below. Find a + b + c. (The grid lines are one unit apart.)    
The parabola passes through (-2,8), (0,0), and (2,8).    

The problem references a graph, but there is no graph provided.    

Taking into consideration only the three points that are specified,    

one parabola that goes through those three points is y = 2x² so    

a+b+c for that curve would be 2 + 0 + 0 = 2      

_.   

Let p be a prime. What are the possible remainders when p is divided by 17?  Select all that apply.    

        17/17 = 1 R 0      

      103/17 = 6 R 1        73/17 = 4 R 5        43/17 = 2 R 9          47/17 = 2 R 13    

        19/17 = 1 R 2        23/17 = 1 R 6        61/17 = 3 R 10        31/17 = 1 R 14    

        37/17 = 2 R 3        41/17 = 2 R 7        79/17 = 4 R 11         83/17 = 4 R 15    

        89/17 = 5 R 4        59/17 = 3 R 8        29/17 = 1 R 12         67/17 = 3 R 16    

So, we can say, shown empirically, that all the possible     

remainders are every integer from 0 to 16, inclusive.    

_.    

Let x and y be integers.  If x and y satisfy 37x + 82y = -172 and x + y = -1, then find x.    

                                            37x + 82y = –172    

                                                x   +   y =     –1    ==>>    y = –x – 1     

substitute (–x – 1) for y   

in the other equation    

                                            37x + 82(–x – 1)  =  –172    

                                            37x – 82x – 82     =  –172    

                                                              – 45x  =  – 90     

                                                                      x  =  2    

_.   

Find the sum of all values of $k$ for which $x^2 + kx - 9x + 25 - 9$ is the square of a binomial.  

combine like terms and   

arrange as ax² + bx + c                              x² + (k – 9) • x + 16    

For the quadratic to be the square of a binomial, the "b" term    

has to be twice the value of the square root of the "c" term.    

                            k – 9  =  (2)(+4)    ==>>    k  =  ( +8 + 9 )    

                            when  k = 17    ==>>   x² + (17 – 9) • x + 16  =  (x + 4)²    

                            when  k =   1    ==>>   x² + (  1 – 9) • x + 16  =  (x – 4)²    

   sum of both values of  k  =  18       

_.