Find the number of ways of seating 10 students A 1 , A 2 , … , A 10 in a row such that A 1 , A 2 , A 3 sit together.

Suppose, we have 10 places in which we can make students sit. Now, the students A 1 , A 2 , A 3 should always sit together. So, they can sit in first, second, and third place which is denoted by a 1 , a 2 , a 3 (say), or they can sit in second, third, and fourth place i.e., a 2 , a 3 , a 4 ,...., and so on. Thus, A 1 , A 2 , A 3 can occupy the places a 1 , a 2 , a 3 , a 2 , a 3 , a 4 , a 3 , a 4 , a 5 , a 4 , a 5 , a 6 , a 5 , a 6 , a 7 , a 6 , a 7 , a 8 , a 7 , a 8 , a 9 , a 8 , a 9 , a 10 . Thus, A 1 , A 2 , A 3 can sit together in 8 ways. Now, A 1 , A 2 , A 3 can arrange among themselves in 3 ! ways. And, the remaining 7 students can be arranged in 7 ! ways. Thus, the total number of ways of seating 10 students A 1 , A 2 , . . . . , A 10 such that A 1 , A 2 , A 3 sit together = 8 × 7 ! × 3 ! = 8 ! × 3 !

Find the number of ways of seating 10 students A 1 , A 2 , … , A 10 in a row such that A 1 , A 2 , A 3 sit in a specified order.

Suppose, we have 10 places in which we can make students sit. Now, we need to find the number of ways of seating 10 students A 1 , A 2 , A 3 , . . . , A 10 in a row such that A 1 , A 2 , A 3 sit in a specified order. Now, A 1 , A 2 , A 3 can occupy any of the 3 places out of 10 places. But there places are fixed so they cannot be arranged among themselves. Hence, the number of ways in which A 1 , A 2 , A 3 can sit = C 3 10 Now, the remaining 7 students can occupy the remaining 7 places and this can be done in 7 ! ways. Thus, the total number of ways in which A 1 , A 2 , A 3 sit in a specified order = C 3 10 × 7 ! = 10 ! 3 ! · 7 ! × 7 ! = 10 × 9 × 8 × 7 × 6 × 5 × 4 = 604 , 800 ways

Find the number of ways of seating 10 students A 1 , A 2 , … , A 10 in a row such that A 1 , A 2 , A 3 sit together in a specified order.

Suppose, we have 10 places in which we can make students sit. Now, the students A 1 , A 2 , A 3 should always sit together. So, they can sit in first, second, and third place which is denoted by a 1 , a 2 , a 3 (say), or they can sit in second, third, and fourth place i.e., a 2 , a 3 , a 4 ,...., and so on. Thus, A 1 , A 2 , A 3 can occupy the places a 1 , a 2 , a 3 , a 2 , a 3 , a 4 , a 3 , a 4 , a 5 , a 4 , a 5 , a 6 , a 5 , a 6 , a 7 , a 6 , a 7 , a 8 , a 7 , a 8 , a 9 , a 8 , a 9 , a 10 . Thus, A 1 , A 2 , A 3 can sit together in 8 ways. Now, A 1 , A 2 , A 3 cannot arrange among themselves as they sit together in a specified order. And, the remaining 7 students can be arranged in 7 ! ways. Thus, the total number of ways of seating 10 students A 1 , A 2 , . . . . , A 10 such that A 1 , A 2 , A 3 sit together in a specified order = 8 × 7 ! = 8 ! = 40 , 320 ways.