SME
Discrete Mathematics · Problem type
Words with repeated letters
Use this for “How many different words can be made from all 11 letters of MISSISSIPPI?”
This is an example. Type your own values to start a new problem.
W
=
!
!
×
!
×
!
×
!
Formulas
W
=
n
!
n₁
! ×
n₂
! ×
n₃
! ×
n₄
!
34,650
=
11!
4! × 4! × 2! × 1!
Assumptions
•
Every letter is used once in each word; copies of one letter look the same.
•
Swapping the k copies of one letter gives the same word, so divide n! by k! for each letter.
•
Type the copies of up to four letters that repeat; a letter used once divides by 1! = 1.
Step-by-step
WE KNOW
n = 11, n₁ = 4, n₂ = 4, n₃ = 2, n₄ = 1
FIND
different words (W)
Step 1 · Find different words (W)
Use W = n! ÷ (n₁! × n₂! × n₃! × n₄!)
Count
the
n
!
orders
as
if
every
letter
were
different,
then
divide
out
the
swaps
of
each
letter’s
copies.
11!
4! × 4! × 2! × 1!
=
39,916,800
24 × 24 × 2 × 1
39,916,800
1,152
=
34,650
W
=
34,650
Check
Put the numbers back in. Both sides should match.
34,650
=
11!
4! × 4! × 2! × 1!
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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