Find population parameter from probability

7 A bag contains \(2 m\) yellow and \(m\) green counters. Three counters are chosen at random, without replacement. The probability that exactly two of the three counters are yellow is \(\frac { 28 } { 55 }\). Determine the value of \(m\).

6. Sixteen cards have been lost from a pack, which therefore contains only 36 cards. Two cards are drawn at random from the pack. The probability that both cards are red is \(\frac { 1 } { 3 }\).

1. Show that \(r\), the number of red cards in the pack, satisfies the equation $$r ( r - 1 ) = 420$$
2. Hence or otherwise find the value of \(r\).
3. Find the probability that, when three cards are drawn at random from the pack,
   1. at least two are red,
   2. the first one is red given that at least two are red.

7. A bag contains \(n\) marbles of which 7 are green. From the bag, 3 marbles are selected at random.
The random variable \(X\) represents the number of green marbles selected.
The cumulative distribution function of \(X\) is given by

1. Show that \(n ( n - 1 ) ( n - 2 ) = 7980\)
2. Verify that \(n = 21\) satisfies the equation in part (a). Given that \(n = 21\)
3. find the exact value of \(a\) and the exact value of \(b\)
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