Determine the number of distinct arrangements of the word APPOINTMENT if the letters MINT must be together.
Please be quick
Answers
Answer:
Answer down
Step-by-step explanation:
First, we can ask ourselves "How many arrangements can we make with 7 things (letters in this case)"
The answer is
7
!
(We can choose 7 in the first slot, 6 in the next, etc)
However, we must also take into account the repetition of some letters in the word STREETS.
We can do this by dividing the repetition.
The repetition occurs because we counted S in place of another S as a possible combination. However, this is not distinct.
What is wrong: "
S
T
R
E
1
E
2
T
S
" is one combination and so is
S
T
R
E
2
E
1
T
S
What is right:
S
T
R
E
1
E
2
T
S
" and
S
T
R
E
2
E
1
T
S
are not distinct, so we only count them once.
We divide the possible arrangements for EACH repeating letter.
Since there are 3 repeating letters: S twice, T twice, E twice,
we have
7
!
2
!
2
!
2
!
Divide it all out and we have 630 distinct combinations
Note: for these types of problems, we can simply do
total letters! / number of repeats of letter1 ! number of repeats of letter2 ! ... and so on for all repeating letters
Answer:
110
Step-by-step explanation:
The formula for a combination of choosing r ways from n possibilities is:
nPr = n!
(n - r)!
where n is the number of items and r is the number of arrangements.
Plugging in our numbers of n = 11 and r = 2 into the permutation formula:
11P2 = 11!
(11 - 2)!
Remember from our factorial lesson that n! = n * (n - 1) * (n - 2) * .... * 2 * 1
Calculate the numerator n!:
n! = 11!
11! = 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
11! = 39,916,800
Calculate the denominator (n - r)!:
(n - r)! = (11 - 2)!
(11 - 2)! = 9!
9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
9! = 362,880
Calculate our permutation value nPr for n = 11 and r = 2:
11P2 = 39,916,800
362,880
11P2 = 110