how many five-digit integers can one make from the digits 2,2,6,0,9,4? I want to ask two questions:

(i) How many five-digit integers can one make from the digits $2, 2, 6, 0, 9, 4$?
(ii) How many digits in (i) satisfy that there is no $6$ followed by $2$?

My Approach
(i) I divided it into cases

*

*if we have $2$ in the first place so we have $1 \times 5 \times 4 \times 3 \times 2$ ways to choose the other 4 digits.


*if we have $2$ in the second place so we have $3 \times 1 \times 4 \times 3 \times 2$ ways to choose the 5 digits.


*if we have only one 2 so we have $3 \times 4 \times 4 \times 3 \times 2$ way to choose
and the result is the sum of the 3 cases.
(ii) I considered 62 as one digit, but couldn't solve it.

There seem to be some issues in your reasoning. The first question can be solved by considering three scenarios:


*

*The integer contains one $2$. There are four possibilities for the first digit (it cannot be $0$), after which the remaining four digits need to be assigned. The number of possibilities equals: $$4 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 96$$

*The integer contains two $2$s and no $0$. We choose the positions of the $2$s and then assign the positions of the remaining digits: $${5 \choose 2} \cdot 3 \cdot 2 \cdot 1 = 60$$

*The integer contains two $2$s and a $0$. We choose the position of the $0$, then choose the positions of the two $2$s, then choose the two remaining digits and finally assign their positions: $${4 \choose 1} \cdot {4 \choose 2} \cdot {3 \choose 2} \cdot 2 = 144$$
The total number of valid integers dus equals:
$$96 + 60 + 144 = 300$$
To answer the second question, we can consider all cases where the combination $62$ occurs. Using the same distinction as before:


*

*$62$ in front with $3 \cdot 2 \cdot 1 = 6$ possibilities or $62$ in second, third or fourth position, so assign $62$, then $0$, then the remaining digits for a total of $3 \cdot 2 \cdot 2 \cdot 1 = 12$ possibilities

*Consider $62$ as one of four digits, for a total of $4 \cdot 3 \cdot 2 \cdot 1 = 24$ possibilities

*$62$ in front with $3 \cdot {2 \choose 1} \cdot 2 \cdot 1 = 12$ possibilities or $62$ in second, third or fourth position, so assign $62$, then $0$, then the remaining digits for a total of $3 \cdot 2 \cdot {2 \choose 1} \cdot 2 \cdot 1 = 24$ possibilities


The number of possible integers thus equals:
$$300 - 6 - 12 - 24 - 12 - 24 = 222$$
