Question

The number of integers greater than$6000$ that can be formed using the digits $3,5,6,7,8$ without repetition is?

• A. $72$
• B. $120$
• C. $192$
• D. $216$

Answer 1

The correct option is C

$192$

The given digits are $3,5,6,7,8$.

Since there are five digits, We can form four and five-digit numbers greater than $6000$.

Step 1: Four digit numbers greater than$6000$

For the number to be greater than $6000$, The first place can be filled with digits greater than or equal to $6$.

Therefore, the first place can be filled in $3$ ways.

The remaining three places can be filled in ${}^{4}P_3=4\times 3\times 2=24$ ways.

Therefore, the total number of four-digit numbers that can be formed using the digits $3,5,6,7,8$ such that it is greater than $6000$ is $24\times 3=72$.

Step 2: Five-digit numbers greater than $6000$

All the five-digit numbers are greater than $6000$, therefore we can place any digit at any place.

We can fill these five places with the given five digits in $5!$ ways

Therefore, the number of five-digit numbers that can be formed using the digits $3,5,6,7,8$ such that it is greater than $6000$ is $5!=5\times 4\times 3\times 2\times 1=120$.

Step 3: The total number of integers greater than $6000$

The total number of integers greater than $6000=$ The number of four-digit numbers $+$The number of five-digit numbers

$$\begin{gathered} =72+120 \\ =192 \end{gathered}$$

$\therefore$The number of integers greater than$6000$ that can be formed using the digits $3,5,6,7,8$ without repetition is $192$.

Hence, the correct option is (C).