Question

Of the 20 members of a cricket club 2 are wicket keepers and 5 bowlers. We have to select a team of 11 from this lot if (a) the team is to include only 1 wicket keeper and at least 3 bowler? (b) if 2 wicket keepers, 5 and 3 bowlers are included?

Answer is - 54054

Plz show how to do it

Answer 1

See below:

Explanation:

We have a cricket club with 20 members: 2 wicket keepers, 5 bowlers, and 13 general members.

With picking a team, we're looking at combination calculations:

`C_(n,k)=((n),(k))=(n!)/((k!)(n-k)!)` with `n="population", k="picks"`

a

We want 1 wicket keeper and at least 3 bowlers.

To do this, let's first look at the wicket keeper. There are 2 of them and we want 1, which gives:

`((2),(1))=2`

We want at least 3 bowlers. Let's pick 3 and then if more show up in the final pick, that's fine:

`((5),(3))=10`

And now the rest of the team. We don't want the other wicket keeper, so he's excluded. We've also already picked 4 people so they're excluded. We need 7 people to round out the team and there are 15 members left we can pick from `(20-1-4)`:

`((15),(7))=6435`

And so we have:

`2xx10xx6435="128,700"`

b

This part of the question isn't clear, so I'll answer the question as if it's asking for exactly 3 of the 5 bowlers are picked.

We first need to pick both wicket keepers:

`((2),(2))=1`

And exactly 3 bowlers:

`((5),(3))=10`

What changes, however, is that now we don't want to pick the final 7 players from 15 possible players (as we did in a) because we could end up with more bowlers than asked for. So instead we pick from the 13 general members:

`((13),(7))=1716`

Giving:

`1xx10xx1716="17,160"`