**Q-How many four-digit numbers can be formed from the digits 1, 3, 4, 5, 6 and 7, if repetition of the digit is not allowed?… and if the numbers are greater than 5,000?**

### Solution

**We have 6 digits available.**

There are 3** ways** to fill the **first digit** (**5,6,7**)

**5 ways** to fill the **second digit**

**4 ways** to fill the **third digit**

**3 ways** to fill the **fourth digit**

3×5×4×3=180

**180, four-digit numbers if the numbers are greater than 5,000 can be formed from the digits 1, 3, 4, 5, 6 and 7 where the digits are used repetition is not allowed**

**Q-2 **How many 3 digits numbers can be formed from the digits 1,2,3,4,5 and 6 if repetition of the digit is allowed and if repetition of the digit is not allowed?

**The first digit can be selected in six ways; and another digit can also be selected in six ways**

**Including the third digit increases the number of ways to **

36 \times 6= 216.

**Note: If repetition is not allowed, the number of ways will be**

6 \times 5 \times 4 = 120

**Q3- How many numbers can be formed from 1, 2, 3, 4, 5 if repetition of the digit is not allowed**

Single digit | 5= \;5\;ways |

Two digit | 5\times4=\;20\; ways |

three digit | 5\times4\times 3= \;60\; ways |

four digit | 5\times4\times 3\times2 =\;120\; ways |

five digits | 5\times4\times 3\times2\times1= \;120\; ways |

**the number of total numbers = 5 + 20 + 60 + 120 + 120=325**