m \u003d k 1 + k 2 + ... + k n , in which the given elements a 1 , a 2 , …, a n are repeated respectively k 1 , k 2 , .., k n times.
Theorem. The number of different permutations with repetitions from the elements {a 1 , …, an} , in which the elements a 1 , …, an are repeated k 1 , ..., kn times, respectively, is equal to
Theorem. The number of different permutations with repetitions from the elements {a 1 , …, an} , in which the elements a 1 , …, an are repeated k 1 , ..., kn times, respectively, is equal to
Words and phrases with rearranged letters are called anagrams. How many anagrams can you make from the word macaque?
Solution.
There are 6 letters in the word "MACAKA" (m=6) .
Determine how many times each letter is used in a word:
"M" - 1 time ( k 1 \u003d 1)
"A" - 3 times (k 2 \u003d 3)
"K" - 2 times (k 3 \u003d 2)
P =
m!
k1 ! _ k2 ! _ … k n !
R 1,3,2 =
6 !
1 ! 3 ! 2 !
=
4*5*6
2
=
60 .
check yourself
1) How many different words can be obtained by rearranging the letters of the word "mathematics" ?
SOLUTION
SOLUTION
check yourself
1) How many different words can be obtained by rearranging the letters of the word "mathematics" ?
Solution.
There are 10 letters in the word "MATHEMATICS" (m=10) .
Let's determine how many times each letter is used in the word: "M" - 2; "A" - 3; "T" - 2; "E" - 1; "I" - 1; "K" -1. (k 1 , k 2 , … , k n )
R 2,3,2,1,1,1 =
10 !
2! 3! 2! 1! 1! 1!
=
151200.
4*5*6*7*8*9*10
2*2
=
check yourself
2) In how many ways can a set of white pieces (a king, a queen, two rooks, two bishops and two knights) be placed on the first horizontal of the chessboard?
SOLUTION
