Permutations And Combinations

Required number of ways

We know,

Answer (b)

Out of consonants, the number of ways of selecting consonants
Similarly, number of ways of selecting
vowels out of vowels
Total number of words formed

Step 1: Recall the formula for permutations. The number of permutations of objects taken at a time is given by .
Step 2: Expand the permutation terms in the given equation. The left-hand side of the equation is: The right-hand side of the equation is . So, we have .
Step 3: Use permutation properties to simplify the left-hand side. We use the property . However, the coefficient of the second term is 5, not 4. Let's try another approach: This is incorrect. The relation is , so . Let's use the definition directly: This still doesn't look like . Consider the identity . So . Then . Now, we need . If , . If , . There seems to be an issue. Let me try to use the property . Let's consider the relationship . If , . There is likely a mistake in my application of properties or a subtlety I am missing. Revisiting the calculation: . If , . Let me try to work backwards from the correct answer (C) . If , . We need , which is .

Step 1: Write down the given expression.
We need to evaluate: This is a sum of combinations:
Step 2: Use the hockey-stick (Christmas stocking) identity.
The hockey-stick identity states that for integers , : Rewrite the sum :
For , ,
For , .
So the sum is: This is . Using the hockey-stick identity with , from 47 to 51:
Step 3: Combine with the first term.
The original expression is:
Step 4: Verify with a numerical check.
For , compute directly:
,
Sum: , , , , ,
Total sum = ,
Expression = ,
, which matches.

Step 5: Select the correct answer.
The expression simplifies to , matching option (C).