Let
x
denote a vector of length q consisting of 0s and 1s. It can be interpreted as an ‘opinion’ comprised of a particular set of responses to a questionnaire consisting of q questions, each having {0, 1}-valued answers. Suppose that the questionnaire is answered by n individuals, thus providing n ‘opinions’. Probabilities of the answer 1 to each question can be, basically, arbitrary and different for different questions. Out of the 2
q
different opinions, what number, μ
n
, would one expect to see in the sample? How many of these opinions, μ
n
(k), will occur exactly k times? In this paper we give an asymptotic expression for μ
n
/ 2
q
and the limit for the ratios μ
n
(k)/μ
n
, when the number of questions q increases along with the sample size n so that n = λ2
q
, where λ is a constant. Let p(
x
) denote the probability of opinion
x
. The key step in proving the asymptotic results as indicated is the asymptotic analysis of the joint behaviour of the intensities np(
x
). For example, one of our results states that, under certain natural conditions, for any z > 0, ∑1
{np(
x
) > z} = d
n
z
−u
,
d
n
= o(2
q
).