How manu four-digit numbers greater than 5000 can be formed from the digits 0,1,2,3,4,5 if
a) all digits may be repeated (I get 215, I just want to confirm if I am right)
b) only the digit 4 may be repeated?
How manu four-digit numbers greater than 5000 can be formed from the digits 0,1,2,3,4,5 if
a) all digits may be repeated (I get 215, I just want to confirm if I am right)
b) only the digit 4 may be repeated?
Hello, azuki!
The first digit must be 5: .$\displaystyle 5\:\_\:\_\:\_$How many four-digit numbers greater than 5000
can be formed from the digits 0,1,2,3,4,5 if
b) only the digit 4 may be repeated?
There are 4 cases to consider . . .
(1) No 4's
Then we have a choice of {0, 1, 2, 3}.
. . The second digit has 4 choices.
. . The third digit has 3 choices.
. . The fourth digit has 2 choices.
There are: .$\displaystyle 4\cdot3\cdot2 \:=\:{\color{blue}24}$ numbers with no 4's.
(2) One 4
There are 3 possible positions for the 4.
. . The other two digits have: $\displaystyle 4\cdot3 \:=\:12$ choices.
There are: .$\displaystyle 3\cdot12 \:=\:{\color{blue}36}$ numbers with one 4.
(3) Two 4's
There are 3 possible positions for the two 4's.
. . There are $\displaystyle 4$ choices for the fourth digit.
There are: .$\displaystyle 3\cdot4\:=\:{\color{blue}12}$ numbers with two 4's.
(4) Three 4's.
There is only $\displaystyle {\color{blue}1}$ number with three 4's: .$\displaystyle 5444$
Answer: .$\displaystyle 24+ 36 + 12 + 1 \:=\:73$ numbers.