How do you find the number of distinct arrangements of the letters in BOOKKEEPER?

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drumette824ed · Accepted Answer

Step 1There are a total of 10 letters.If they were all distinguishable then the number of distinct arrangements would be 10!. We can make them distinguishable by adding subscripts:      B          O      1              O      2              K      1              K      2              E      1              E      2        P          E      3        R  If we remove the subscripts from the letter O&#039;s, then it no longer makes any difference what order the O&#039;s are in and we find that             1              2        !              =          1      2       of our 10! arrangements are identical to the other half.So there are                     10        !                    2        !             possible arrangements of the letters:      B    O    O          K      1              K      2              E      1              E      2        P          E      3        R  Step 2If we remove the subscripts from the letter K&#039;s a similar thing happens and we are left with half again. So there are                     10        !                    2        !        2        !             possible arrangements of the letters:      B    O    O    K    K          E      1              E      2        P          E      3        R  Finally, since             E      1      ,             E      2       and             E      3       can be arranged in 3! possible orders, then when we remove the subscripts from the E&#039;s there are                     10        !                    2        !        2        !        3        !             distinct arrangements of the letters:BOOKKEEPER                    10        !                    2        !        2        !        3        !              =                  10        !                    2        ⋅        2        ⋅        6              =                  10        !                    4        !              =    10    ⋅    9    ⋅    8    ⋅    7    ⋅    6    ⋅    5    =    151200

How do you find the number of distinct arrangements of

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