We may have (3 men and 2 women) or (4 men and 1 woman) or (5 men only).

So, Required number of ways = = (⁷C₃ x ⁶C₂) + (⁷C₄ x ⁶C₁) + (⁷C₅)

=       \(\left(\frac{7\times6\times5}{3\times2\times1}\times\frac{6\times5}{2\times1}\right)\) + (⁷C₃ x ⁶C₁) + (⁷C₂)

= \(525+\left(\frac{7\times6\times5}{3\times2\times1}\times6\right)+\left(\frac{7\times6}{2\times1}\right)\)

= (525 + 210 + 21)

= 756.

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.

So,, Required number of ways = (120 x 6) = 720.

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 =  \(\frac{7!}{2!}=2520.\)

Now, 5 vowels in which O occurs 3 times and the rest are different, can be arranged

in  \(\frac{5!}{3!}=20ways.\)

So, Required number of ways = (2520 x 20) = 50400.

Number of ways of selecting (3 consonants out of 7) and (2 vowels out of 4)

= (⁷C₃ x ⁴C₂)

= \(\left(\frac{7\times6\times5}{4\times3\times1}\times\frac{4\times3}{2\times1}\right)\)

= 210.

Number of groups, each having 3 consonants and 2 vowels = 2

Number of ways of arranging 
5 letters among themselves   = 5!

= 5 x 4 x 3 x 2 x 1

= 120.

So,Required number of ways = (210 x 120) = 25200.