In the card game Red gamma, the deck consists of 49 cards, with cards numbered 1

Question

In the card game Red gamma, the deck consists of 49 cards, with cards numbered 1 through 7 in each of the seven colors of the rainbow (red, orange, yellow, green, blue, indigo, violet). All 49 cards are distinct. You are dealt a hand of seven cards. Answer each question and briefly explain your reasoning. Leave your answer unsimplified in terms of permutations, combinations, exponents, etc. (a) How many seven-card hands are possible? (b) How many seven-card hands contain no blue cards? (c) How many seven-card hands contain all cards of the same color? (d) How many seven-card hands contain one card of each color? (e) How many seven-card hands contain at least one red card?

Explanation / Answer

Given number of cards: 49.
Cards of each color: 7.
a. How many seven-card hands are possible?
Selecting 7 cards out of 49 is:
   C(49, 7) = 85,900,584.
b. How many seven-card hands contain no blue cards?
   So, hands without blue cards is:
   C(42, 7) = 26,978,328.
c. How many seven-card hands contain all cards of the same color?
   Obviously 7, as there are 7 colors.
d. How many seven-card hands contain one card of each color?
   7 power 7 = 13,841,287,201.

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