Pat writes all the 7-digit numbers in which all the digits are different and each digit is greater than the one to its right (so the tens digit is greater than the units, the hundreds greater than the tens, and so on). For example, 9,865,320 is one of the numbers that Pat writes down.

(a) How many numbers does Pat write down?

(b) One of Pat's numbers is chosen at random. What is the probability that the tens digit is a 1?

(c) One of Pat's numbers is chosen at random. What is the probability that the middle (thousands) digit is a 5?

Note: of course, you could solve this problem by repeating Pat's experiment and writing down all of the numbers. But don't do that -- figure out the answers without needing to write down all of the numbers!

Guest Apr 3, 2015

#5**+36 **

**Can I make this a shorter answer?** **YES I CAN**

Pat writes all the 7-digit numbers in which all the digits are different and each digit is greater than the one to its right (so the tens digit is greater than the units, the hundreds greater than the tens, and so on). For example, 9,865,320 is one of the numbers that Pat writes down.

(a) How many numbers does Pat write down?

9,8,7,6,5,4,3,2,1,0 3 of the 10 must be removed 10C3 = 120

(b) One of Pat's numbers is chosen at random. What is the probability that the tens digit is a 1?

(9,8,7,6,5,4,3,2) 1,0 3 of the 8 must be left out 8C3= 56

P(1 in the 10s column) = $${\frac{{\mathtt{56}}}{{\mathtt{120}}}} = {\frac{{\mathtt{7}}}{{\mathtt{15}}}} = {\mathtt{0.466\: \!666\: \!666\: \!666\: \!666\: \!7}}$$

(c) One of Pat's numbers is chosen at random. What is the probability that the middle (thousands) digit is a 5?

(9,8,7,6 one of these four must be left out) 5 ( 4,3,2,1,0 two iof thse 5 must be left out) 4C1*5C2=4*10=40

P(5 is in the middle) = $${\frac{{\mathtt{40}}}{{\mathtt{120}}}} = {\frac{{\mathtt{1}}}{{\mathtt{3}}}} = {\mathtt{0.333\: \!333\: \!333\: \!333\: \!333\: \!3}}$$

Melody
Apr 6, 2015

#2**+12 **

Thanks Alan,

This looks interesting but finding the time to answer all these counting and probability questions (especially in full) is becoming very difficult. I think I speak for more answerers than just myself

If you want more explanation try asking for it with a new post.

Please follow these instructions :)

Melody
Apr 4, 2015

#3**+12 **

I agree. That's why I cheated on this one and just wrote a simple program to do the calculations!

.

Alan
Apr 4, 2015

#4**+20 **

Thanks Anon,

My instructions for reposting were followed to the T so I am going to give some time to your question.

REPOST: http://web2.0calc.com/questions/why-no-explanation-i-wasn-t-the-one-who-posted-this-twice

Alan has already give the answer only version so that will help me know if my answers are correct. Thanks Alan.

---------------

**PLEASE NOTE THAT I HAVE DONE IT A MUCH SHORTER WAY IN THE NEXT POST.**

Pat writes all the 7-digit numbers in which all the digits are different and each digit is greater than the one to its right (so the tens digit is greater than the units, the hundreds greater than the tens, and so on). For example, 9,865,320 is one of the numbers that Pat writes down.

(a) How many numbers does Pat write down?

First I will look at all those beginning in 9

9 8 7 6 5 4 3 Ends in 3 5 middle digits 0 digits can be left out 5C0=1

9 2 Ends in 2 6 middle digits any 1 digit must be left out 6C1=6

9 1 Ends in 1 7 middle digits any 2 digit must be left out 7C2=21

9 0 Ends in 0 8 middle digits any 3 digit must be left out 8C3=56

Now look at those starting with 8

8 7 6 5 4 3 2 Ends in 2 5 middle digits 0 digits can be left out 5C0=1

8 1 Ends in 1 6 middle digits any 1 digit must be left out 6C1=6

8 0 Ends in 0 7 middle digits any 2 digit must be left out 7C2=21

Now look at those starting with 7

7 6 5 4 3 2 1 Ends in 1 5 middle digits 0 digits can be left out 5C0=1

7 0 Ends in 0 6 middle digits any 1 digit must be left out 6C1=6

Now look at those starting with 6

6 5 4 3 2 1 0 Ends in 0 5 middle digits 0 digits can be left out 5C0=1

$${\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6}}{\mathtt{\,\small\textbf+\,}}{\mathtt{21}}{\mathtt{\,\small\textbf+\,}}{\mathtt{56}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6}}{\mathtt{\,\small\textbf+\,}}{\mathtt{21}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}} = {\mathtt{120}}$$

I agree with Alan, I must be a genious!

**(b) One of Pat's numbers is chosen at random. What is the probability that the tens digit is a 1?**

Mmm

9 - - - - 10 (8,7,6,5,4,3,2 ) 3 out of the 7 will be missing 7C3 = 35

8 - - - - 10 (7,6,5,4,3,2) 2 out of the 6 will be missing 6C2 = 15

7 - - - - 10 (6,5,4,3,2) 1 out of the 5 will be missing 5C1 = 5

6 5 4 3 2 1 0 0 out of the 4 will be missing 4C0 = 1

**$${\mathtt{35}}{\mathtt{\,\small\textbf+\,}}{\mathtt{15}}{\mathtt{\,\small\textbf+\,}}{\mathtt{5}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}} = {\mathtt{56}}$$ **

56 of the numbers will have a 1 in the 10s column.

P(1 in the 10s column)** = $${\frac{{\mathtt{56}}}{{\mathtt{120}}}} = {\frac{{\mathtt{7}}}{{\mathtt{15}}}} = {\mathtt{0.466\: \!666\: \!666\: \!666\: \!666\: \!7}}$$**

WOW that the same as Alan got too - I'm on a roll.

(c) One of Pat's numbers is chosen at random. What is the probability that the middle (thousands) digit is a 5?

NOTE: I have put the 5 in the MIDDLE which is the 10,000 place

_ _ _ 5 _ _ _

9,8,7,6 leave out 1 of these 4C1= 4 ways

4,3,2,1,0 leave out 2 of these 5C2 = 10 ways

4*10 = 40 ways that 5 can be in the middle

P(5 is in the middle) = $${\frac{{\mathtt{40}}}{{\mathtt{120}}}} = {\frac{{\mathtt{1}}}{{\mathtt{3}}}} = {\mathtt{0.333\: \!333\: \!333\: \!333\: \!333\: \!3}}$$

This one does not agree with Alan. Maybe I am not so clever after all.

-----------------------------------------------------------------------

I am thinking that I may have made this question longer than necessary.

**I might try and make the working shorter in another post. **

Melody
Apr 6, 2015

#5**+36 **

Best AnswerPat writes all the 7-digit numbers in which all the digits are different and each digit is greater than the one to its right (so the tens digit is greater than the units, the hundreds greater than the tens, and so on). For example, 9,865,320 is one of the numbers that Pat writes down.

(c) One of Pat's numbers is chosen at random. What is the probability that the middle (thousands) digit is a 5?

**Can I make this a shorter answer?** **YES I CAN**

(a) How many numbers does Pat write down?

9,8,7,6,5,4,3,2,1,0 3 of the 10 must be removed 10C3 = 120

(b) One of Pat's numbers is chosen at random. What is the probability that the tens digit is a 1?

(9,8,7,6,5,4,3,2) 1,0 3 of the 8 must be left out 8C3= 56

P(1 in the 10s column) = $${\frac{{\mathtt{56}}}{{\mathtt{120}}}} = {\frac{{\mathtt{7}}}{{\mathtt{15}}}} = {\mathtt{0.466\: \!666\: \!666\: \!666\: \!666\: \!7}}$$

(9,8,7,6 one of these four must be left out) 5 ( 4,3,2,1,0 two iof thse 5 must be left out) 4C1*5C2=4*10=40

P(5 is in the middle) = $${\frac{{\mathtt{40}}}{{\mathtt{120}}}} = {\frac{{\mathtt{1}}}{{\mathtt{3}}}} = {\mathtt{0.333\: \!333\: \!333\: \!333\: \!333\: \!3}}$$

Melody
Apr 6, 2015

#6**+11 **

You are right about the probability of the thousands digit equalling 5 being 1/3 Melody. I just checked my program and realised I'd put the counter in the wrong place!

.

Alan
Apr 6, 2015

#8**+11 **

The thousands column *is* the middle column!

1 2 3 **4** 5 6 7

units tens hundreds **thousands** ten-thousands hundred-thousands millions

.

Alan
Apr 6, 2015