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A hand consists of 55 cards from a well-shuffled deck of 5252 cards.\newlinea. Find the total number of possible 55-card poker hands.\newlineb. A heart flush is a 55-card hand consisting of all heart cards. Find the number of possible heart flushes.\newlinec. Find the probability of being dealt a heart flush.\newlinea. There are a total of □\square poker hands.

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Q. A hand consists of 55 cards from a well-shuffled deck of 5252 cards.\newlinea. Find the total number of possible 55-card poker hands.\newlineb. A heart flush is a 55-card hand consisting of all heart cards. Find the number of possible heart flushes.\newlinec. Find the probability of being dealt a heart flush.\newlinea. There are a total of □\square poker hands.
  1. Calculate Combination Formula: To find the total number of possible 55-card poker hands, we use the combination formula which is given by C(n,k)=n!k!(n−k)!C(n, k) = \frac{n!}{k!(n - k)!}, where nn is the total number of items to choose from, kk is the number of items to choose, and “!” denotes factorial.\newlineIn this case, n=52n = 52 (total number of cards) and k=5k = 5 (number of cards in a hand).\newlineSo, we calculate C(52,5)=52!5!(52−5)!=52!5!47!C(52, 5) = \frac{52!}{5!(52 - 5)!} = \frac{52!}{5!47!}.
  2. Perform Calculation: Now we perform the calculation for 52!/(5!47!)52! / (5!47!). \newline52!/(5!47!)=(52×51×50×49×48)/(5×4×3×2×1)52! / (5!47!) = (52 \times 51 \times 50 \times 49 \times 48) / (5 \times 4 \times 3 \times 2 \times 1) since the terms from 47!47! will cancel out with the same terms in 52!52!.\newlineThis simplifies to (52×51×50×49×48)/(120)(52 \times 51 \times 50 \times 49 \times 48) / (120).
  3. Calculate Total Poker Hands: We calculate the simplified expression (52×51×50×49×48)/(120)(52 \times 51 \times 50 \times 49 \times 48) / (120).(52 \times 51 \times 50 \times 49 \times 48) / (120) = 311,875,200 / 120 = 2,598,960\. So, there are a total of \$2,598,960 different 55-card poker hands.
  4. Calculate Heart Flushes: To find the number of possible heart flushes, we need to calculate the number of ways to choose 55 cards from the 1313 hearts in the deck.\newlineWe use the combination formula again with n=13n = 13 (total number of heart cards) and k=5k = 5 (number of cards in a heart flush hand).\newlineSo, we calculate C(13,5)=13!5!8!.C(13, 5) = \frac{13!}{5!8!}.
  5. Perform Calculation: Now we perform the calculation for 13!/(5!8!)13! / (5!8!). \newline13!/(5!8!)=(13×12×11×10×9)/(5×4×3×2×1)13! / (5!8!) = (13 \times 12 \times 11 \times 10 \times 9) / (5 \times 4 \times 3 \times 2 \times 1) since the terms from 8!8! will cancel out with the same terms in 13!13!. \newlineThis simplifies to (13×12×11×10×9)/(120)(13 \times 12 \times 11 \times 10 \times 9) / (120).
  6. Calculate Total Heart Flushes: We calculate the simplified expression (13×12×11×10×9)/(120)(13 \times 12 \times 11 \times 10 \times 9) / (120).(13 \times 12 \times 11 \times 10 \times 9) / (120) = 154,440 / 120 = 1,287\.So, there are a total of 1,2871,287 different heart flushes possible.
  7. Calculate Probability: To find the probability of being dealt a heart flush, we divide the number of heart flushes by the total number of 55-card poker hands.Probability=Number of heart flushesTotal number of 5-card poker hands=1,2872,598,960\text{Probability} = \frac{\text{Number of heart flushes}}{\text{Total number of 5-card poker hands}} = \frac{1,287}{2,598,960}.
  8. Calculate Probability: We calculate the probability of being dealt a heart flush.\newlineProbability = 1,2872,598,960≈0.000495\frac{1,287}{2,598,960} \approx 0.000495 (rounded to six decimal places).

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