Four horses are competing in race , and any no. of ties are possible.
Let's consider race positions as:
I II III IV
4 { Here all 4 finishing in Ist place} => # ways = 4C4 = 1
3 1 # ways = 4C3 * 1C1 = 4
4C3 for selecting 3 horses for first place.
2 2 # ways = 4C2 * 2C2 = 6
4C2 for selecting 2 horses for Ist place
& 2C2 for II place
2 1 1 Similarly # ways = 4C2 * 2C1 * 1C1 = 12
1 3 # ways = 4C1 * 3C3 = 4
1 2 1 # ways = 4C1 * 3C2 * 1C1 = 12
1 1 2 # ways = 4C1 * 3C1 * 2C2 = 12
1 1 1 1 # ways = 4C1 * 3C1 * 2C1 * 1C1 = 24
Adding up all these cases , we get
Total # ways = 1 + 4 + 6 + 12 + 4 + 12 + 12 + 24
= 75 ways..