The value of C1 + 3C3 + 5C5 + 7C7 + ...., where C0, C3, C5, C7,..... are binomial coefficients isa)n.2n -1b)n.2n +1c)n.2nd)n.2n-2Correct answer is option 'D'. Can you explain this answer?

Question

Answer

The given expression involves binomial coefficients, which are calculated using the formula C(n, r) = n! / (r!(n-r)!), where n is the total number of items and r is the number of items chosen at a time.

To understand the given expression, let's break it down step by step:

Step 1: C1
C1 represents the binomial coefficient when n = 1 and r = 1. Plugging these values into the formula, we get C(1, 1) = 1! / (1!(1-1)!) = 1.

Step 2: 3C3
3C3 represents the binomial coefficient when n = 3 and r = 3. Plugging these values into the formula, we get C(3, 3) = 3! / (3!(3-3)!) = 1.

Step 3: 5C5
5C5 represents the binomial coefficient when n = 5 and r = 5. Plugging these values into the formula, we get C(5, 5) = 5! / (5!(5-5)!) = 1.

Step 4: 7C7
7C7 represents the binomial coefficient when n = 7 and r = 7. Plugging these values into the formula, we get C(7, 7) = 7! / (7!(7-7)!) = 1.

From the above steps, we can observe that the value of each term in the given expression is 1.

Now, let's generalize the pattern to find the value of the expression for any given value of n:

Step 5: C2n
C2n represents the binomial coefficient when n = 2n and r = 2n. Plugging these values into the formula, we get C(2n, 2n) = (2n)! / ((2n)!((2n)-(2n))!) = 1.

Therefore, for any value of n, the value of the expression C1 3C3 5C5 7C7 ... is always 1.

Hence, the correct answer is option 'D' - n.2n-2.

Answer

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