Calculate your expected winnings (or losses) if you throw away all but the 5♠, 6♠. With this hand you have throw away the 4♦, J♦ and K♣. You might get a pair, 2 pair, 3-of-a-kind, 4-of-a-kind, full-house, straight, flush, straight-flush or nothing. Let R be the set of cards with face value {2,3,7,8,9,10,Q,A} and S be the remaining cards with face value {4, J, K}

pair of jacks or pair of kings: 5♠, 6♠ and one other card from the remaining 38

pair of queens or aces: 5♠, 6♠ and also one other card from the remaining 37

pair of 5s (incl 5♠) and pair of 6s (incl 6♠) and one card from the remaining 41

two pair: pair of 5s (incl 5♠), the 6♠ and a pair from R

two pair: pair of 5s (incl 5♠), the 6♠ and a pair from S

two pair: pair of 6s (incl 6♠), the 5♠ and a pair from R

two pair: pair of 6s (incl 6♠), the 5♠ and a pair from S

3-of-a-kind from R and the 5♠, 6♠

3-of-a-kind from S and the 5♠, 6♠

3-of-a-kind: three 5s (incl 5♠), the 6♠, and one card from the remaining 41

3-of-a-kind: three 6s (incl 6♠), the 5♠, and one card from the remaining 41

4-of-a-kind: four 5s (incl 5♠) and the 6♠

4-of-a-kind: four 6s (incl 6♠) and the 5♠

full house: three 5s (incl 5♠) and two 6s (incl 6♠)

full house: three 6s (incl 6♠) and two 5s (incl 5♠)

straight 6 high or 7 high or 8 high (don't include straight flushes)

straight 9 high (don't include the straight flush)

flush: 3 more cards from the remaining 11 spades (don't include the 4 possible straight flushes)

straight flush: 6 high, 7 high, 8 high or 9 high

nothing can be found by subtracting the total number of hands above from C(47,3) and you should compare this to:

three different cards from R

two different cards from R, one from S

one card from R, two different from S

three different cards from S

subtract off the straights, flushes and straight flushes (because these are included in 1. through 4. and shouldn't be)

a pair of 4s and one card from the remaining 38

a pair of {2,3,7,8,9,10} and one card from the remaining 37

a pair of 5s (incl 5♠) and two different cards from R

a pair of 5s (incl 5♠) and one card from S and one from R

a pair of 5s (incl 5♠) and two different cards from S

a pair of 6s (incl 6♠) and two different cards from R

a pair of 6s (incl 6♠) and one card from S and one from R

a pair of 6s (incl 6♠) and two different cards from S