The number of arrangements of five letters from the word 'magnetic' that contain the word net is:
a)60 b)100 c)360 d)630 e)720
we're still doing this at school and I still don't get it; since I have to use three of the eight letters I only have five left and can only use two of them, so should I multiply 8 by 5?
treat "net" as if it were one item
eg. let X="net"
so now you are simply arranging the X plus 2 of the remaining 5 letters.
Put the X first, then the other two places can be filled in 5x4 ways, which is 20 ways
But the X, the "net" could also be in the middle, or at the end, so there are 3 ways for the X to go
the number of ways is then 3x5x4 = 60
or
Since the X (the "net") now makes your letters spell MAGXIC, which is 5 letters, and we have to fill 3places, the number of ways = 5x4x3 = 60
what is the question asking specifically?
how many possible combinations there were and thank you very much for your explanation :)
oh, okay, wow, so that's the metohd; I get it now; thanks!
before I had no idea how to even set up the equation, but I'm strating to see the pattern!
:D
I get where the five is coming from, since that's the remainder of letters, but where are you getting the four from?
let's pretend we put the X down first, which is really the "net" occupying 3 letter positions
That leaves 2 more positions to be filled from the remaining 5 letters, the MAGIC.
So there are 5 ways to fill the next position, right?
Since we must have picked one of those 5 letters, that would leave 4 letters remaining for the next and last spot.
Hence the 5x4, I multiplied by 3 because the "net" could have been first, in the middle or at the end
Hope that makes sense
e.g. Suppose I want to make all possible arrangements consisting of 4 letters, those 4 letters coming from the word COMPUTER, with no letter appearing more than once
no. of arrangements = 8x7x6x5 = 1680
oh, okay!
thanks a million! :)
To find the number of arrangements of five letters from the word 'magnetic' that contain the word 'net,' we can break down the problem into smaller steps.
Step 1: Count the total number of arrangements of five letters from the word 'magnetic.'
In this case, we have eight letters to choose from ('m', 'a', 'g', 'n', 'e', 't', 'i', 'c'), and we need to select five of them. We can use the concept of combinations to calculate this.
The number of combinations of selecting r items from a set of n items is given by the formula nCr = n! / (r!(n-r)!), where '!' denotes factorial. Applying this formula, we get:
8C5 = 8! / (5!(8-5)!)
= 8! / (5!3!)
= (8 * 7 * 6 * 5!) / (5! * 3 * 2 * 1)
= 8 * 7 * 6 / (3 * 2 * 1)
= 56
So, there are 56 possible arrangements of five letters from the word 'magnetic.'
Step 2: Determine the number of arrangements that contain the word 'net.'
Now we need to figure out how many of those 56 arrangements actually include the word 'net.' To do this, we consider the word 'net' as a single entity.
Since 'net' has three distinct characters ('n', 'e', 't'), we can treat it as a single entity and calculate the number of arrangements of 'magnetic' (minus 'net'). This means we now have only five distinct entities to arrange ('m', 'a', 'g', 'i', 'c').
The number of arrangements of five distinct entities is given by 5!, which is the factorial of 5.
5! = 5 * 4 * 3 * 2 * 1 = 120
So, there are 120 arrangements of 'magnetic' that do not contain the word 'net.'
Step 3: Calculate the number of arrangements that contain the word 'net.'
By subtracting the number of arrangements that do not contain 'net' (120) from the total number of arrangements (56), we can find the number of arrangements that do contain 'net.'
Total arrangements that contain 'net' = Total arrangements - Arrangements without 'net'
= 56 - 120
= -64
Uh-oh! It seems we made a mistake in our calculations, as we ended up with a negative number. This indicates that there might have been an error in our reasoning or calculations along the way.
Let's back up and reconsider the problem:
The number of arrangements that contain the word 'net' can be determined by thinking about the different positions of 'net' within the word 'magnetic.'
Since 'net' has three distinct letters, there are three possible positions for 'net' within each arrangement. We can calculate the number of arrangements for each position and then sum them up.
Position 1: 'net' is in the first three positions (___, ___, ___).
In this case, we have three fixed slots for 'net' and five remaining letters ('m', 'a', 'g', 'i', 'c'). We need to arrange these five letters in their remaining slots.
The number of arrangements for this case is given by 5! = 5 * 4 * 3 * 2 * 1 = 120 arrangements.
Position 2: 'net' is in the middle three positions (___, ___, ___).
Here again, there are three fixed slots for 'net,' and we need to arrange the remaining five letters. The number of arrangements is also 5!.
Position 3: 'net' is in the last three positions (___, ___, ___).
Like the previous cases, we have three fixed slots for 'net,' and the remaining five letters can be arranged in 5! ways.
By adding up the arrangements for each position, we get:
Total arrangements = 120 (Position 1) + 120 (Position 2) + 120 (Position 3) = 360
Therefore, the correct answer is c) 360: The number of arrangements of five letters from the word 'magnetic' that contain the word 'net' is 360.
This problem is a permutation problem. The hint is given in the word "arrangement." However, you could make this to a combination problem by multiplying by r! (since nCr=(nPr)/r! and multiplying it to r! would give you nPr). To do this problem, you have how many can you choose from? (that would be your n) and how many do you choose? (that would be your r)
8 C 3 (8 choose 3)
To enter it in the graphing calculator, you enter 8, cPr, then 3. Changing it into permutation, multiply the result by 3!.