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?

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- Feb 8th 2009, 06:02 AMazuki[SOLVED] Permutation and Combination Question 1
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? - Feb 8th 2009, 06:10 AMPlato
- Feb 8th 2009, 09:08 AMSoroban
Hello, azuki!

Quote:

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.