Answer: Option C

Explanation:
Total number of balls = 9 + 3 + 4
Two balls can be drawn from 16 balls in ¹⁶C₂ ways.

Answer: Option B

Explanation:
The word contains five consonants. Three vowels, three consonants can be selected from five consonants in ⁵C₂ ways, two vowels can be selected from three vowels in ³C₂ ways. 
3 consonants and 2 vowels can be selected in ⁵C₂ . ³C₂ ways i.e., 10 x 3 = 30 ways.

Answer: Option C

Explanation:

We can select one boy from 20 boys in 20 ways.
We select one girl from 25 girls in 25 ways
We select a boy and girl in 20 x 25 ways i.e., = 500 ways.

Answer: Option B

Explanation:
Given that, the question paper consists of five problems. For each problem, one or two or three or none of the choices can be attempted. Hence, the required number of ways = 4⁵ - 1. 
= 2¹⁰ - 1 = 1024 - 1 = 1023

Answer: Option B

Explanation:
The group contain six men and seven women 
Three men can be selected from six men in  ⁶C₃ways.
Four women can be selected from seven women in  ⁷C₄ways.
Total number of ways = (⁷C₄)(⁶C₃).

Answer: Option C

Explanation:  

Since each ring consists of six different letters,
the total number of attempts possible with the three rings is
= 6 * 6 * 6 = 216. Of these attempts, one of them is a successful attempt.
Maximum number of unsuccessful attempts = 216 - 1 = 215.

Answer: Option A

Explanation:
The given digits are six.
The number of four digit numbers that can be formed using six digits is ⁶P₄ = 6 x 5 x 4 x 3 = 360.

Answer: Option D

Explanation:
The given digits are 1, 2, 3, 5, 7, 9
A number is even when its units digit is even. Of the given digits, two is the only even digit.
Units place is filled with only '2' and the remaining three places can be filled in ⁵P₃ ways.
Number of even numbers = ⁵P₃ = 60.

Answer: Option B

Explanation:
Six members can be selected from ten members in
¹⁰C₆ = ¹⁰C₄ ways.

Answer: Option A

Explanation:
In a group of 6 boys and 4 girls, four children are to be selected such that
at least one boy should be there.
Hence we have 4 choices as given below
We can select 4 boys ------(Option 1). 
Number of ways to this = 6_C4
We can select 3 boys and 1 girl ------(Option 2)
Number of ways to this = 6_C3 x 4C1
We can select 2 boys and 2 girls ------(Option 3)
Number of ways to this = 6_C2 x 4_C2
We can select 1 boy and 3 girls ------(Option 4)
Number of ways to this = 6_C1 x 4_C3
Total number of ways 
= (6_C4) + (6_C3 x 4_C1) + (6_C2 x 4_C2) + (6_C1 x 4_C3)
= (6_C2) + (6_C3 x 4_C1) + (6_C2 x 4_C2) + (6_C1 x 4_C1) [Applied the formula n_Cr = n_{C(n - r)} ] 
=[(6×5)/(2×1)]+[(6×5×4)/(3×2×1)×4]+[(6×5)/(2×1)(4×3)/(2×1)]+[6×4]
= 15 + 80 + 90 + 24 = 209