MrWhite
From a class of 20 students whose names are listed in alphabetical order, a teacher will choose one group of 3 students to represent the class in a student congress. If the teacher will not choose a group of 3 students whose names are in 3 consecutive positions on the list, how many different groups of 3 students could be chosen by the teacher?
(A) 1120
(B) 1122
(C) 1135
(D) 1137
(E) 1140
Let's assume that the names of 20 students appear on the list as shown below. \(S_n\) represents the name of the \(n^{\text{th}}\) student
\(S_1 \quad S_2 \quad S_3 \quad S_4 \quad S_5 \quad S_6 \quad S_7 .................... \quad S_{17} \quad S_{18} \quad S_{19} \quad S_{20}\)
If there were no restrictions, the teacher could have chosen the three students in \(^{20}C_3\) ways. However, the teacher cannot choose 3 consecutive positions on the list. Let's try to partition the list such that we can select three consecutive students between the partition.
| | → denotes the partition
\(| S_1 \quad S_2 \quad S_3| \quad S_4 \quad S_5 \quad S_6 \quad S_7 .................... \quad S_{17} \quad S_{18} \quad S_{19} \quad S_{20}\)
\(S_1 \quad |S_2 \quad S_3 \quad S_4| \quad S_5 \quad S_6 \quad S_7 .................... \quad S_{17} \quad S_{18} \quad S_{19} \quad S_{20}\)
\(S_1 \quad S_2 \quad |S_3 \quad S_4 \quad S_5 | \quad S_6 \quad S_7 .................... \quad S_{17} \quad S_{18} \quad S_{19} \quad S_{20}\)
.
.
\(S_1 \quad S_2 \quad S_3 \quad S_4 \quad S_5 \quad S_6 \quad S_7 .................... \quad | S_{17} \quad S_{18} \quad S_{19} | \quad S_{20}\)
\(S_1 \quad S_2 \quad S_3 \quad S_4 \quad S_5 \quad S_6 \quad S_7 .................... \quad S_{17} \quad | S_{18} \quad S_{19} \quad S_{20} |\)
There are 18 ways the students can be partitioned such that three consecutive students appear; we need to subtract this number from \(^{20}C_3\).
Note: We don't have to perform the partition as represented above. I have done so to represent the ways the partition will occur. We can simply count the start or the end positions of the partition to determine the number. For example, the first partition ends after \(S_3\), and the last partition ends after \(S_{20}\)
Number of partitions = \((20 - 3) + 1 = 18\)
Total possible ways = \(^{20}C_3 - 18 = 1140 - 18 = 1122\)
Option B