Permutation, Combination and Probability (UPSC) Part-II

For Task 1 no. of ways = 2
Task 2 no. of ways = 1
Task 3 no. of ways = 3
Task 4 no. of ways = 3
Task 5 no. of ways = 3
Total no. of ways for condition = 3 + 3 + 3 + 2 + 1 = 12

Condition II
When task T₂ is given to be person 4

No. of ways for Task T₁ = 2
No. of ways for Task T₂ = 1
No. of ways for Task T₃ = 3
No. of ways for Task T₄ = 3
No. of ways for Task T₅ = 3
Total number of ways for condition II
= 3 + 3 + 3 + 2 + 1 = 12

Total number of ways for condition I and II = 12 + 12 = 24

Let x be the number of women.
Let y be the number of men.
Total number of hand shakes = xy = 24
Then, the possible factors of x and y are x = 6 or 4, y = 4 or 6

Number of hugs = ⁶C₂ + ⁴C₂
= (6×5)/(2×1) + (4×3)/2
= 15 + 6 = 21

White Marbles      Red Marbles
         10                                13
         White Marbles      Green Marbles
         10 . 0                       5

Now, total number of Marbles = 5 + 10 + 13 = 28

Total number of ways = ²C₁ + ⁵C₂ = 2 + 5×4/2×1 = 2 + 10 = 12

Number of possible combinations of selectors
= 2 × 10 = 20

If mathematics I is not opted, then two subjects out of four subjects have to be opted for.
∴ Number of ways in which two subjects can be opted for = 4×3/2 = 6

If mathematics II is opted, then it can be offered only if mathematics I is also opted for number of ways in which two subjects can be opted for = 6 + 1 = 7.

Let number of a pairs of brown socks = y
Price of brown socks = x
Price of black socks = 3x
According to question ⇒ 5 × 3x + yx = 100 ... (i)
Now, clerk has interchanged socks pairs then price is increased by 100%

3xy + 5x = (15x + yx) + (15x + yx) × 100 / 100
⇒ 3xy + 5x = 30x + 2xy
⇒ 30x + 2xy = 3xy + 5x ⇒ 25x = xy ⇒ y = 25
∴ So, number of brown socks = 25

— B₁ — B₂ —
2 boys can take their seats in 2! ways and 3 girls can take the remaining 3 seats in ³C₂ × 2! ways.

Hence, 2! × ³C₂ × 2! = 12 ways.