Q1: Consider the word BASKETBALL
a) how many of the arrangements begin with a k? (60 480)
b) how many of the arrangements start with a B and end with a K? (10 080)
For this question, I am not sure what they mean but it beginning with k or b...
Q2: Joanne's bag of marbles contain one red, three blue and four green marbles. If she reaches in to select some without looking, how many different arrangements could she make? (39)
I feel that I have the most trouble understanding the question so I don't know how to start...
you have
BB
AA
LL
E
T
K
S
start with K
= 1x9!/(2!2!2!) = 45360
( I think you divided by 3 times 2!)
b) BxxxxxxxxK
number = 8!/(2!2!)
= 10080
(you had that one right, the other B is now just one of the guys)
Q2: A bit tricky.
e.g. she could take them all, she could take 3 greens and 2 blue or .....
there is one red
she can take the red in 2 ways:
- don't take it, take 1
there are 3 blues
she can take the blues in 4 ways,
- don't take it, take 1, take 2, or take all three
there are 4 green
she can take the greens in 5 ways,
- dont't take it, take 1, .....
Number of ways to take marbles
= 2(4)(5) = 40 ways
but that includes the case of not taking a red, not taking a blue and not taking a green.
It says she took "some" so we will subtract that 1 case
Number of ways to select marbles = 40-1 = 39
@reiny thank you!
A1: To answer question 1, we need to find the number of arrangements of the word "BASKETBALL" that meet the given conditions.
a) To find the number of arrangements that begin with a "K," we need to fix the position of "K" at the beginning of the word and rearrange the remaining letters. We can treat the remaining letters (B, A, S, E, T, B, A, L, L) as distinct entities.
To calculate this, we use the formula for the number of arrangements of distinct objects, which is n!, where n is the number of distinct objects. In this case, there are 9 distinct objects (9 letters excluding the repeated "B"), so the number of arrangements starting with a "K" is 9!.
However, there are two "B" letters, so we need to divide by 2! to account for their indistinguishability. Therefore, the number of arrangements starting with a "K" is 9! / 2!.
Plugging the values into a calculator, we find the answer to be 60,480.
b) To find the number of arrangements that start with a "B" and end with a "K," we fix the positions of "B" and "K" at the beginning and end of the word, respectively. Similarly, we treat the remaining letters as distinct entities.
So, we have the letters "B," "A," "S," "E," "T," "B," "A" for rearrangement. There are 7 distinct objects, so the number of arrangements starting with a "B" and ending with a "K" is 7!.
Again, we have two "B" letters, so we divide by 2! to account for their indistinguishability. Therefore, the number of arrangements starting with a "B" and ending with a "K" is 7! / 2!.
Calculating this value, we find the answer to be 10,080.
A2: To answer question 2, we need to find the number of different arrangements Joanne can make when selecting marbles from her bag without looking.
Joanne's bag contains one red, three blue, and four green marbles. When she reaches in to select marbles, the order in which she selects them matters (arrangements).
To find the number of arrangements, we add up the number of ways Joanne can select each color of marble without regard to the order within each color.
She can select the red marble in 1 way, the blue marbles in 3! = 6 ways (3 factorial since there are 3 blue marbles), and the green marbles in 4! = 24 ways (4 factorial since there are 4 green marbles).
To calculate the total number of arrangements, we multiply these numbers together: 1 * 6 * 24 = 144.
Therefore, Joanne can make 144 different arrangements when selecting marbles from her bag without looking.