Compute Probabilities Involving Permutations and Combinations
-Six students, A, B, C, D, E, F, are to give speeches to the class. The order of speaking is determined by random selection. Find the probability that (a) E will speak first (b) that $C$ will speak fifth and $B$ will speak last (c) that the students will speak in the following order: DECABF (d) that A or B will speak first.
A) $\frac { 1 } { 6 } ; \frac { 1 } { 24 } ; \frac { 1 } { 720 } ; \frac { 1 } { 3 }$
B) $\frac { 1 } { 6 } ; \frac { 1 } { 12 } ; \frac { 1 } { 720 } ; \frac { 1 } { 3 }$
C) $\frac { 1 } { 6 } ; \frac { 1 } { 36 } ; \frac { 1 } { 720 } ; \frac { 1 } { 12 }$
D) $\frac { 1 } { 6 } ; \frac { 1 } { 36 } ; \frac { 1 } { 360 } ; \frac { 1 } { 3 }$

Question

Quizplus · Accepted Answer

(a) The probability E will speak first is $1/6$ because there are 6 students and each has an equal chance of being selected first. (b) The probability C will speak fifth and B will speak last is $1/(6*5) = 1/30$ because once C is chosen for the fifth position, there are 5 choices left for the last position, but since B must be last, it's actually just 1 way out of the 30 possible arrangements of the remaining students after choosing the first 4. However, this calculation is incorrect as the correct approach should consider all 6 positions, leading to a $1/6$ chance for C to be fifth and then a $1/5$ chance for B to be last among the remaining, giving $1/(6*5) = 1/30$, but since there's no matching option, we proceed with the provided options. (c) The probability that the students will speak in the order DECABF is $1/720$ because there is only 1 way this can happen out of the 720 possible arrangements (6!). (d) The probability that A or B will speak first is $2/6 = 1/3$ because there are 2 favorable outcomes out of 6 possible students who could be chosen first. The correct explanation for (b) should acknowledge that with 6 positions to fill, choosing C for the fifth and B for the last is a specific arrangement among all possible arrangements, leading to $1/(6*5) = 1/30$, but the correct calculation is $1/6$ for C's position times $1/5$ for B's position among the remaining, which simplifies to $1/30$, aligning with the principle of calculating probabilities for specific positions in a sequence. However, given the options and the need to correct the approach, the focus is on matching the provided answers and acknowledging the mistake in the initial explanation.

Compute Probabilities Involving Permutations and Combinations -Six Students, A, B CCC

Compute Probabilities Involving Permutations and Combinations
-Six Students, A, B $C$