How many six digit number contains exactly 4 different digit?
Any help would be greatly appreciated?
Thanks,
Hello, a69356!
I will assume that the number may begin with zero.How many six-digit numbers contains exactly 4 different digits?
First, select 4 digits from the available 10 digits.
. . There are: .$\displaystyle {10\choose4}\:=\:{\color{blue}210}$ choices.
Then there are two possible distributions of the digits.
. . . $\displaystyle \begin{array}{cc}(1)& \{A,A,A,B,C,D\} \\ \\[-4mm]
(2) & \{A,A,B,B,C,D\} \end{array}$
Case (1): .$\displaystyle A,A,A,B,C,D$
. . We have a Triple and three Singletons.
There are 4 choices for the Triple
Then the letters can be arranged in: $\displaystyle {6\choose3,1,1,1} = {\color{blue}120} $ ways.
There are: .$\displaystyle 210\cdot4\cdot120 \:=\:110,\!800\text{ Case-one numbers.}$
Case (2): .$\displaystyle A,A,B,B,C,D$
. . We have two Pairs and two Singletons.
There are: $\displaystyle {4\choose2} \:=\:{\color{blue}6}$ choices for the two Pairs.
Then the letters can be arranged in: $\displaystyle {6\choose2,2,1,1} \:=\:{\color{blue}180}$ ways.
There are: .$\displaystyle 210\cdot6\cdot180 \:=\:226,800 \text{ Case-two numbers.}$
Therefore, there are: .$\displaystyle 100,\!800 + 226,\!800 \;=\;\boxed{327,\!600}$ six-digit numbers
. . that contain four different digits.
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If leading zeros are not allowed, consider this fact:
. . One-tenth of the above numbers begin with zero.