Combinations

To solve the given problem, we need to evaluate the expression:

This expression is composed of two parts:

1. Evaluate the First Part:

The first part can be represented as:

Now, consider the binomial expansion:

Taking the derivative with respect to in the expansion and setting gives:

Therefore,

Setting :

2. Evaluate the Second Part:

Using the combinatorial identity for the sum of binomial coefficients at odd indices:

Set and :

and

Thus, and .

Adding the two equations, the sum of coefficients at odd positions:

Final Calculation:

Combine both parts:

The original expression also states which is included in .

Hence, the final answer is:

Correct Answer:

The correct answer is (A) :

First we arrange 5 red cubes in a row and assume x₁, x₂, x₃, x₄, x₅ and x₆ number of blue cubes between them
Here we see,
x₁ + x₂ + x₃ + x₄ + x₅ + x₆ = 11
also,
x₂, x₃, x₄, x₅ ≥ 2
Hence,
x₁ + x₂ + x₃ + x₄ + x₅ + x₆ = 3
Therefore ,
Number of solutions found = ⁸C₅
= 56

The correct answer is 180
Factors of 36 = 2²⋅ 3²⋅ 1
Five-digit combinations can be
(1, 2, 2, 3, 3) (1, 4, 3, 3, 1), (1, 9, 2, 2, 1)
(1, 4, 9, 11) (1, 2, 3, 6, 1) (1, 6, 6, 1, 1)
i.e., total numbers

= (30 × 3) + 20 + 60 + 10 = 180

Hence, the correct option is (B):

Determining Numbers Divisible by 15

To determine numbers divisible by 15, the following conditions must be met:

• The last digit must be 5 (a condition for divisibility by 5).
• The sum of the digits must be divisible by 3 (a condition for divisibility by 3).

Possible Combinations

Explanation:

For each combination of digits, the number of possible arrangements (permutations) that satisfy the conditions is determined. For example, the combination 1215 has 3 valid numbers since the arrangement must end with 5 and satisfy divisibility by 3.

Similar calculations are performed for all other combinations.

Total Numbers:

Total Numbers = 3 + 3 + 3 + 3 + 6 + 3 = 21