The number of dots given are 1, 3, 6, 10, 15, ...

1^st number = 1
2^nd number = 1 + 2 = 3
3^rd number = 3 + 3 = 6
4^th number = 6 + 4 = 10
5^th number = 10 + 5 = 15
6^th number will be 15 + 6 = 21
7^th number will be 21 + 7 = 28
8^th number will be 28 + 8 = 36
9^th number will be 36 + 9 = 45
10^th number will be 45 + 10 = 55
Number of dots in the 10^th figure = 55.

(iii) Every part A, B, C, D, E, F, G, H, I, J, K, L, M, N, O and P are 16 triangles.
Combining two at a time. A & B, B & C, C & D, D & A, J & K, L & M, N & O, P & I are 8 triangles.
Taking the combination of 4, we have I, J, E & A; K, L, F and B, M, N, G and C, P, O, H and D are 4 triangles.
Taking 8 parts together we have QRS, RST, STQ, and TRQ are 4 triangles.
\(\therefore \) Total triangles = 16 + 8 + 4 + 4 = 32 triangles.

A, B, C, D, & E are five triangles.
No combination of these triangles taken 2 at a time, form any triangle.
Only the combinations of A, F & C; A, F & D; B, F & E; B, F & D and C, F & E form triangles when taken 3 triangles at a time.
Hence, the total number of triangles = 5 + 5 = 10.

A, B, C, D are four triangles.
Combining A & D, B & D and D & C do not form any triangles.
Combining A, B & D; A, C & D and B, D & C also do not form any triangles.
Combining A, B, C & D = 1 triangle
\(\therefore\)Total number of triangles = 4 +1 = 5

We have given two shorts which are black and blue colour. Take it as T black and T  blue.
Also we have 3 shirts, coloured white, blue and red denoted by S white, S blue and S red.
Now fix T black and then T blue the different combinations are

Thus we get total of 6 combinations as
Black short and White shirt
Black short and Blue shirt
Black short and Red shirt
Blue short and White shirt
Blue short and Blue shirt
Blue short and Red shirt.
But it is given short and shirt are of different colours.
\(\therefore\) We give up Blue short and Blue shirt combination. So we have 5 different combinations.