Four digit numbers with all digits distinct are formed using the digits $1,2,3,4,5,6,7$ in all possible ways…
Four digit numbers with all digits distinct are formed using the digits $1,2,3,4,5,6,7$ in all possible ways. If $p$ is the total number of numbers thus formed and $q$ is the number of numbers greater than 3400 among them, then $p: q=$
$3: 2$
$4: 3$
$6: 5$
$7: 4$
Solution
$p=$ Total four digit number's formed using $1,2,3,4$,
$5,6,7={ }^7 C_4 \times 4!=\frac{7!4!}{3!4!}=840$
For number greater than 3400 ,
Numbers starting with 3,4 are $={ }^5 C_2 \times 2!=20$
Now, first digit can be $4,5,6,7={ }^4 C_1 \times{ }^6 C_3 \times 3!=480$
Numbers starting with 3 and second digit 5, 6, 7 are
$={ }^3 C_1 \times{ }^5 C_2 \times 2!=\frac{3!\times 5!}{3!2!}=60$
$\therefore q=$ Total numbers greater than $3400=20+480+60=560$ Now, $p: q=840: 560=3: 2$.