Question

The results of 21 football matches (win, lose, draw) are to be predicted. The no. of different forecasts that can contain 19 wins is:
A. 210
B. 640
C. 840
D. 1260

Answer 1

Hint: Form the cases based on the given condition that out of 21 matches played by the team 19 matches are won. After making the cases, find the number of possible ways that event or case can occur. And in the end add all the values obtained in each of the cases.


Complete step-by-step solution
Let us begin with the fact that out of 21 football matches played by the team, 19 matches have been won by the team. Thus, we can observe that there can be three cases to find the number of different forecasts that contain 19 wins.
Those cases can be listed as:
1. There are 19 wins and 2 losses.
2. There are 19 wins, 1 loss and 1 draw.
3. There are 19 wins and 2 draws.

Let us consider these cases one by one and find out their possible outcomes.

Case 1:- There are 19 wins and 2 losses.
The number of possible ways is given by dividing 21! by the possibility 19 wins, that is 19! multiplied by the possibility of 2 losses, that is 2!
$
   \Rightarrow N = \dfrac{{21!}}{{19! \times 2!}} \\
   = \dfrac{{21 \times 20}}{2} \\
   = 210 \\
$

Case 2:- There are 19 wins, 1 loses and 1 draw.
The number of possible ways is given by dividing 21! by the possibility 19 wins, that is 19! multiplied by the possibility of 1 lose, that is 1! and the possibility of 1 draw, that is, 1!.
$
   \Rightarrow N = \dfrac{{21!}}{{19! \times 1! \times 1!}} \\
   = \dfrac{{21 \times 20}}{1} \\
   = 420 \\
$

Case 3:- There are 19 wins and 2 draws.
The number of possible ways is given by dividing 21! by the possibility 19 wins, that is 19! multiplied by the possibility of 2 draws, that is 2!
$
   \Rightarrow N = \dfrac{{21!}}{{19! \times 2!}} \\
   = \dfrac{{21 \times 20}}{2} \\
   = 210 \\
$


Thus, now we add all the possible ways obtained in all the 3 cases to get the no. of different forecasts that can contain 19 wins.
$
   \Rightarrow {\text{Total}} = 210 + 420 + 210 \\
   = 840 \\
$

Hence, option (C) is the correct option.

Note: Listing down all the possible ways is a cumbersome and time taking task, thus we use the method of permutation or fundamentals of counting principles to solve questions like these. Make sure to analyze everything properly to not skip any case. Also, make your concept very clear about where to apply the repetition method and where not to. Also, do not mix combinations with permutation. Do not make calculation errors.