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Correct
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1
'n' identical objects in 'r' distinct slots
$$^{n+r-1}C_{r-1}$$
2
'n' identical objects in 'r' distinct slots, at least 1 in each slot
$$^{n-1}C_{r-1}$$
3
'n' distinct objects in 'r' distinct slots
$$r^n$$
4
'n' distinct objects in 'r' distinct slots, arrangement in the slot matters
$$\frac{(n+r-1)!}{(r-1)!}$$
5
No. of positive integral solutions of $$x_1+x_2+x_3+.....+x_n=s$$
$$^{s-1}C_{n-1}$$
6
No. of non-negative integral solutions of $$x_1+x_2+x_3+.....+x_n=s$$
$$^{s+n-1}C_{n-1}$$
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