Of the Death Stars personnel 7 people are from Coruscant 14

Of the Death Star’s personnel, 7 people are from Coruscant, 14 from Alderaan, 4 from Jakku, 5 from Korriban, 2 from Mandalore, and 3 from Bespin. A committee of 6 people is to be formed to meet with Lord Vader to fail him for the last time.
How many committees with at least 2 representatives from Alderaan are possible?

Solution

Given , People from Coruscant = 7

People from Alderaan = 14

People from Jakku = 4

People from Korriban = 5

People from Mandalore = 2

People from Bespin =3

we have to form a committee of 6 people and in that we need at least 2 representatives from Alderaan.

atleast 2 means >=2 which implies we can take 2 or 3 or 4 or 5 or 6 representatives from Alderaan.

case 1 : if we take 2 representatives from Alderaan in 14_C2 ways.

then we have to take rest of 4 representatives from the rest of 21(7+4+5+2+3) representatives

(Coruscant,Jakku,Korriban,Mandalore,Bespin) in 21_C4 ways.

so total number of committees in the case is = 14_{C2 *} 21_C4 = 91 * 5985 = 544635

(from the fromula n_Cr = n! / (r!*(n-r)!) ).

case 2 : if we take 3 representatives from Alderaan in 14_C3 ways.

then we have to take rest of 3 representatives from the rest of 21(7+4+5+2+3) representatives

(Coruscant,Jakku,Korriban,Mandalore,Bespin) in 21_C3 ways.

so total number of committees in the case is = 14_{C3 *} 21_C3 = 364 * 1330 = 484120

(from the fromula n_Cr = n! / (r!*(n-r)!) ).

case 3 : if we take 4 representatives from Alderaan in 14_C4 ways.

then we have to take rest of 2 representatives from the rest of 21(7+4+5+2+3) representatives

(Coruscant,Jakku,Korriban,Mandalore,Bespin) in 21_C2 ways.

so total number of committees in the case is = 14_{C4 *} 21_C2 = 1001 * 210 = 210210

(from the fromula n_Cr = n! / (r!*(n-r)!) ).

case 4 : if we take 5 representatives from Alderaan in 14_C5 ways.

then we have to take rest of 1 representative from the rest of 21(7+4+5+2+3) representatives

(Coruscant,Jakku,Korriban,Mandalore,Bespin) in 21_C1 ways.

so total number of committees in the case is = 14_{C5 *} 21_C1 = 2002 * 21 = 42042

(from the fromula n_Cr = n! / (r!*(n-r)!) ).

case 5 : if we take 6 representatives from Alderaan in 14_C6 ways.

then we have to take rest of 0 representatives from the rest of 21(7+4+5+2+3) representatives

(Coruscant,Jakku,Korriban,Mandalore,Bespin) in 21_C0 ways.

so total number of committees in the case is = 14_{C6 *} 21_C0 = 3003 * 1= 3003 (n_C0 = 1)

(from the fromula n_Cr = n! / (r!*(n-r)!) ).

Total number of committees from all cases = 544635+484120+210210+42042+3003

= 1284010