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I am trying to work on counting skills and came up with the following problem, however I feel that I am running into an issue of double counting or counting faulty cases.

Say there are 9 distinct birds flying in small groups. Any 4 may be leaders and any 5 may be followers, but each leader must have at least ONE follower. What number of combinations of leader/followers exist?

Attempt at solution:using n choose k (NcK)

(9c4) x [(5c2)(3c1)(2c1)(1c1) + (5c1)(4c2)(2c1)(1c1) + (5c1)(4c1)(3c2)(1c1) + (5c1)(4c1)(3c1)(2c2)]

I realize I may be counting cases where a leader(s) is without a follower... though I am not sure how to account for this. Is there a way to account for invalid cases with basic combination methods? (I am assuming there is a mistake here.)

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    Does every bird have to be a leader or a follower? I mean, can there be one leader, one follower and seven normal birds, say? – 2017-01-26
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    All birds must be either leader/follower. – 2017-01-26

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The simplest approach I see is $9c4$ ways to pick the leaders, $4$ to pick which leader gets two followers, $5c2$ to get the two followers, $3!$ to arrange the other three followers among the leaders, so $$9c4 \cdot 4 \cdot 5c2 \cdot 3!=126 \cdot 4 \cdot 10 \cdot 6=30240$$

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    Thank you. My method seemed to return the same value, and I was concerned my method counted cases where leaders had no followers. Your explanation is more intuitive and makes perfect sense. – 2017-01-26