Answer: 1,663,200.
Step-by-step explanation:
In the word "PARALLELEPIPED", there are 12 letters, with repetitions as follows:
'P' appears 2 times
'A' appears 1 time
'R' appears 1 time
'L' appears 3 times
'E' appears 4 times
'I' appears 1 time
'D' appears 1 time
Step 2: Use the Formula for Arrangements with Repeated Elements
Given a word with
�
n letters, where each distinct letter has a certain frequency (repetitions), the number of arrangements can be calculated as:
arrangements
=
�
!
�
1
!
×
�
2
!
×
…
×
�
�
!
arrangements=
f
1
!×f
2
!×…×f
k
!
n!
where
�
1
,
�
2
,
…
,
�
�
f
1
,f
2
,…,f
k
are the frequencies of the distinct letters.
Step 3: Apply the Formula to "PARALLELEPIPED"
Here,
�
=
12
n=12. The frequency counts are:
'P': 2 times
'A': 1 time
'R': 1 time
'L': 3 times
'E': 4 times
'I': 1 time
'D': 1 time
Plugging into the formula, we have:
arrangements
=
12
!
(
2
!
)
×
(
1
!
)
×
(
1
!
)
×
(
3
!
)
×
(
4
!
)
×
(
1
!
)
×
(
1
!
)
.
arrangements=
(2!)×(1!)×(1!)×(3!)×(4!)×(1!)×(1!)
12!
.
Let's break this down step-by-step:
12
!
=
12
×
11
×
10
×
9
×
8
×
7
×
6
×
5
×
4
×
3
×
2
×
1
≈
479001600
12!=12×11×10×9×8×7×6×5×4×3×2×1≈479001600.
2
!
=
2
2!=2.
3
!
=
3
×
2
×
1
=
6
3!=3×2×1=6.
4
!
=
4
×
3
×
2
×
1
=
24
4!=4×3×2×1=24.
Using these values, the arrangements calculation becomes:
arrangements
=
479001600
2
×
6
×
24
=
479001600
288
=
1663200.
arrangements=
2×6×24
479001600
=
288
479001600
=1663200.
Thus, the total number of arrangements for the word "PARALLELEPIPED" is 1,663,200.
