Two cards are drawn without replacement from an ordinary deck of 52 playing cards. What are

the odds against drawing a club and a diamond?

To calculate the odds against drawing a club and a diamond, we first need to determine the number of possible outcomes and the number of favorable outcomes.

Number of possible outcomes:
When two cards are drawn without replacement from a deck of 52 playing cards, the total number of possible outcomes can be calculated using the combination formula: nCr = n! / (r!(n-r)!), where n is the total number of cards (52) and r is the number of cards drawn (2).
Therefore, the number of possible outcomes is 52C2 = 52! / (2!(52-2)!) = 1326.

Number of favorable outcomes:
To draw a club and a diamond, we need to consider the number of clubs and diamonds in the deck.
There are 13 clubs and 13 diamonds in a deck of 52 playing cards.
The probability of drawing a club in the first draw is 13/52 (since there are 13 clubs in a deck of 52 cards).
After drawing a club, there would be 51 remaining cards, including 12 clubs.
The probability of drawing a diamond in the second draw is 13/51 (since there are 13 diamonds remaining in the 51 cards).
Therefore, the number of favorable outcomes is (13/52) * (13/51) = 169/2652 = 1/156.

Odds against drawing a club and a diamond:
Odds against an event represent the ratio of unfavorable outcomes to favorable outcomes.
Therefore, the odds against drawing a club and a diamond can be calculated as (number of possible outcomes - number of favorable outcomes) / number of favorable outcomes.
In this case, the odds against drawing a club and a diamond would be (1326 - 1) / 1 = 1325:1.

So the odds against drawing a club and a diamond are 1325 to 1.