## Explanation:

This is a combinations question.

The number of ways you can combine n objects taken r at a time is given by:

^nC_r=\frac{n!}{r!(n-r)!}

Although there are 52 cards in a deck, only \frac{52}{4}=13 of them are hearts. So in this example n=13 and r=5.

^{13}C_5=\frac{13!}{5!\times(13-5)!}=\frac{13!}{5!\times(8)!} \\ =\frac{13\times12\times11\times10\times9}{5\times4\times3\times2}

=13\times3\times11\times3=1287

So, the number of hands possible consisting of only hearts is 1287.

**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**

**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

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

Total Number of 5 digits Numbers which can be formed by numbers 1,2,3,4,5 (without repeating digits)

= 5\times4\times3\times2 = 5! = 120.

**Q- 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**

**How many 5 digit numbers can be formed using (0-9) 0,1,2,3,4,5,6,7,8,9**

10\times10\times10\times10\times10=10^5=100000

**The number of signals that can be sent by 5 flags of different colors taking one or more at a time is**

Solution

Number of signals using one flag

{\color{Blue} ^{5}\textrm{P}_1}= 5

Number of signals using two flags

{\color{Blue} ^{5}\textrm{P}_2} = 20

Number of signals using three flags

{\color{Blue} ^{5}\textrm{P}_3} = 60

Number of signals using four flags

{\color{Blue} ^{5}\textrm{P}_4} = 120

Number of signals using five flags

{\color{Blue} ^{5}\textrm{P}_5} = 120

Therefore, the total number of signals using one or more flags at a time is

5 + 20 + 60 + 120 + 120= 325

**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 digit4 ways to fill the third digit3 ways to fill the fourth digit**

3×5×4×3=180