This question refers to a standard deck of playing cards. If you are unfamiliar with playing cards, there is an explanation in P

Answer:

The total number of combinations is 2595960

P(two hearts) = 0.003 %

P(hearts and diamonds) = 63.73 %

P(4 same rank) = 0.024 %

P(full house) = 0.144 %

P(No same rank) = 50.7 %

Explanation:

How many unique five-card hands can be created from a standard deck of 52 cards?

There are 52 cards to choose from, and we need to pick 5 cards

The number of combinations equals 52C5

The total combinations = 2595960

Now, how many five-card hands contain precisely two hearts?

We have 13 hearts, and we need to select two of them

P(two hearts) = 13C2/52C5

Resulting in: P(two hearts) = 78/2598960

So, P(two hearts) = 0.003 %

What about five-card hands composed solely of hearts and diamonds?

P(hearts and diamonds) = 13C5*13C5/52C5

P(hearts and diamonds) = 1287*1287/2598960

Thus, P(hearts and diamonds) = 1656369/2598960

This gives P(hearts and diamonds) = 63.73 %

How many five-card hands have four cards of the same rank?

P(4 same rank) = 13C1*12C1*4C1/5C2

So we have P(4 same rank) = 13*12*4/2598960

Consequently, P(4 same rank) = 624/2598960

Resulting in P(4 same rank) = 0.024 %

For five-card hands containing a full house, we have

P(full house) = 13C2*2C1*4C3*4C2/5C2

This results in P(full house) = 78*2*4*6/2598960

Thus P(full house) = 3744/2598960

Giving us P(full house) = 0.144 %

Finally, how many hands have no two cards of the same rank?

P(No same rank) = 13C5*4C1*4C1*4C1*4C1*4C1/5C2

Calculating gives P(No same rank) = 1287*4*4*4*4*4/2598960

Hence P(No same rank) = 1317888/2598960

Leading to P(No same rank) = 50.7 %

Answer:

Explanation:

The number of combinations to select 10 girls is 35C₁₀

The number of combinations to select 10 boys is also 35C₁₀

Thus, the total combinations for selection is 35C₁₀ x 35C₁₀