D

We may have (3 men and 2 women) or (4 men and 1 woman) or (5 men only).
i.e, Required number of ways = [latex](^{7}{C}_{3} \times ^{6}{C}_{2})[/latex] + [latex](^{7}{C}_{4} \times ^{6}{C}_{1})[/latex] + [latex](^{7}{C}_{5})[/latex]
= [latex]\frac {7 \times 6 \times 5}{6 \times 5} \times \frac {6 \times 5 }{2 \times 1}[/latex] + [latex](^{7}{C}_{3} \times ^{6}{C}_{1})[/latex] + [latex](^{7}{C}_{2})[/latex]
= 525 + [latex](\frac {7 \times 6 \times 5}{6 \times 5} \times 6) + (\frac {7 \times 6}{2 \times 1})[/latex]
= (525 + 210 + 21)
= 756.

C

The word 'LEADING' has 7 different letters.
When the vowels EAI are always together, they can be supposed to form one letter.
Then, we have to arrange the letters LNDG (EAI).
Now, 5 (4 + 1 = 5) letters can be arranged in 5! = 120 ways.
The vowels (EAI) can be arranged among themselves in 3! = 6 ways.
Required number of ways = (120 x 6) = 720.

C

In the word 'CORPORATION', we treat the vowels OOAIO as one letter.
Thus, we have CRPRTN (OOAIO).
This has 7 (6 + 1) letters of which R occurs 2 times and the rest are different.
Number of ways arranging these letters = [latex]\frac {7!}{2!}[/latex] = 2520.
Now, 5 vowels in which O occurs 3 times and the rest are different, can be arranged in [latex]\frac {5!}{3!}[/latex] = 20 ways.
i.e, Required number of ways = (2520 x 20) = 50400

C

Number of ways of selecting (3 consonants out of 7) and (2 vowels out of 4)
= [latex](^{7}{C}_{3} \times ^{4}{C}_{2})[/latex]
= [latex](\frac {7 \times 6 \times 5}{3 \times 2 \times 1} \times 6) + (\frac {4 \times 3}{2 \times 1})[/latex]
= 210.
Number of groups, each having 3 consonants and 2 vowels = 210.
Each group contains 5 letters.
Number of ways of arranging
5 letters among themselves = 5!
= 5 x 4 x 3 x 2 x 1
= 120.
Required number of ways = (210 x 120) = 25200.

C

The word 'LEADER' contains 6 letters, namely 1L, 2E, 1A, 1D and 1R.
i.e, Required number of ways = [latex]\frac {6!}{(1!)(2!)(1!)(1!)(1!)}[/latex] = 360