This problem is a combination of the Poisson distribution and binomial distribution.
First, we need to find the probability of a single student sending less than 6 messages in a day, i.e.
P(X<6)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)
=0.006738+0.033690+0.084224+0.140374+0.175467+0.175467
= 0.615961
For ALL 20 students to send less than 6 messages, the probability is
P=C(20,20)*0.615961^20*(1-0.615961)^0
=6.18101*10^(-5) or approximately
=0.00006181
Answer:

Step-by-step explanation:
![We\ are\ given:\\TU=SQ\\TP=PQ\\\angle TPU= \angle SPQ [Vertically\ Opposite\ Angles\ Are\ Equal]\\Hence,\\As\ we\ are\ given,\ 2\ sides\ and\ 1\ angle\ of\ each\ triangle\ correspond,\ we\\ could\ use\ the\ SAS\ Congruency Rule.\\But:\\](https://tex.z-dn.net/?f=We%5C%20are%5C%20given%3A%5C%5CTU%3DSQ%5C%5CTP%3DPQ%5C%5C%5Cangle%20TPU%3D%20%5Cangle%20SPQ%20%5BVertically%5C%20Opposite%5C%20Angles%5C%20Are%5C%20Equal%5D%5C%5CHence%2C%5C%5CAs%5C%20we%5C%20are%5C%20given%2C%5C%202%5C%20sides%5C%20and%5C%201%5C%20angle%5C%20of%5C%20each%5C%20triangle%5C%20correspond%2C%5C%20we%5C%5C%20could%5C%20use%5C%20the%5C%20SAS%5C%20Congruency%20Rule.%5C%5CBut%3A%5C%5C)
<em>As SAS Congruency Rule tells us that 'Two triangles are congruent only if two sides and an included angle of one triangle corresponds to two sides and an included angle of the other' .</em>
<em>Here,</em>
<em>As ∠TPU and ∠SPQ are NOT the included angle of ΔTUP and ΔSPQ respectively, the two triangles cannot be proven congruent through SAS Congruency.</em>
<em>Note: We also cannot apply SSA congruency as SSA congruency doesnt exist.</em>
I think buying 36 ears of corn for $7.20 is the better buy. 6 ears of corn for $1.25 cost .20+ per ear of corn. Buying the 36 count buys them at an even .20. Plus he should have enough for everybody without having to worry about buying more.
If you need a more detail explaination, tell me. :)