There are 52 cards in a standard deck. How many 5-card hands are possible if the cards are all hearts?

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Explanation:

This is a combinations question.

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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)

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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 digit5= \;5\;ways
Two digit5\times4=\;20\; ways
three digit5\times4\times 3= \;60\; ways
four digit5\times4\times 3\times2 =\;120\; ways
five digits5\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 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


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