Latest self Aptitude Question SOLUTION: in a running competition,there were 4 runners.In how many ways, counting ties, can 4 runners cross the finishing line?

in 16 ways

24 ways

It can be solved using 4P4=4*3*2*1=24

hey friends, it is not like that.
Consider the case of four horses. The horses may finish in 1, 2, 3, or 4 blocks. Labeling the horses a, b, c, d, one example of three blocks is: (a) (b) (c, d). This means that horses c and d have tied, and that horses a, b, and (c, d) have finished separately.

Note further that the three blocks -- (a), (b), (c, d) -- may be arranged in 3! = 6 ways. Clearly this is true of any partition that consists of three blocks.

The table below shows, for each number of blocks, the possible partitions and their number, the number of arrangements per partition, and the number of outcomes.
No.of
blocks Partitions No.of partitions Arrangements Outcomes
per partition

1 (a, b, c, d) 1 1! 1
2 (a)(b,c,d),
(b)(a,c,d),
(c)(a,b,d),
(d)(a,b,c),
(a,b)(c,d),
(a,c)(b,d),
(a,d)(b,c) 7 2! 14

3 (a)(b)(c,d),
(a)(c)(b,d),
(a)(d)(b,c),
(b)(c)(a,d),
(b)(d)(a,c),
(c)(d)(a,b) 6 3! 36

4 (a) (b) (c) (d) 1 4! 24

Hence, counting ties, four horses can cross the finishing line in 1 + 14 + 36 + 24 = 75 ways.