There are 5 students in class 10, 6 students in class 11 and 8 students in class 12. If the number of ways in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students of classes 10 and 11 is 100k, then k is equal to .........
Answer: 238
Solutions · 1
Official solution
Let the number selected from classes 10, 11, 12 be (a,b,c) with a≥2,b≥2,c≥2, a+b+c=10, a+b≤5 (so c≥5).
Valid triples: (2,3,5)-type with a+b≤5: enumerate (2,2,6),(2,3,5),(3,2,5):
\[\binom52\binom62\binom86 + \binom52\binom63\binom85 + \binom53\binom62\binom85\]
\[= 10\times15\times28 + 10\times20\times56 + 10\times15\times56\]
=4200+11200+8400=23800
\[100k = 23800 \implies k = 238\]