Arranging the Letters of the word TRIANGLE to form words that start with T, end with R, start with T and end with R
Problem 1
| 1 |
| 8 |
| 1 |
| 8 |
| 1 |
| 56 |
Solution
Number of Letters/Characters
= 8
{T, R, I, A, N, G, L, E}
⇒ nL = 8
Experiment :
Forming a word using the 8 letters
Total Number of Possible Choices
= Number of words that can be formed using the 8 letters
| ⇒ n | = | 8P8 |
| = | 8! |
When there are large factorial values, using the factorial form would reduce the burden of calculations
Let
A : the event of the word formed starting with T
B : the event of the word formed ending with R
C : the event of the word formed starting with T and ending with R
For Event A
Number of letters fixed, each in its own place
= 1
⇒ nFL = 1
After fixing the specified letters in their respective places
Number of letters remaining
= Total Number of letters − Number of letters fixed in specific places
| ⇒ nRL | = | nL − nFL |
| = | 8 − 1 | |
| = | 7 |
Number of places remaining to be filled
= Total Number of Places − Number of letters fixed in specific places
| ⇒ nRP | = | nP − nFL |
| = | 8 − 1 | |
| = | 7 |
Number of Favorable Choices
= Number of words that can be formed using the letters of the word TRIANGLE fixing T in the first place
= Number of ways in which the specified letters can be fixed each in its own place × Number of ways in which the remaining letters can be arranged in the remaining places
= nFLPnFL × nRLPnRP
= 1 × nRLPnRP
= nRLPnRP
| ⇒ mA | = | 7P7 |
| = | 7! |
Probability that the word formed using all the letters of the word TRIANGLE starts with T
⇒ Probability of occurrence of Event A
| = |
|
| ⇒ P(A) | = |
| ||
| = |
| |||
| = |
| |||
| = |
|
Odds
= Total Number of possible choices − Number of Favorable choices
| ⇒ mAc | = | n − mA |
| = | 8! − 7! | |
| = | 8 × 7! − 7! | |
| = | (8 − 1) × 7! | |
| = | 7 × 7! |
in favor
Odds in Favor of the word formed starting with T⇒ Odds in Favor of Event A
= Number of Favorable Choices : Number of Unfavorable Choices
= mA : mAc
= 7! : 7 × 7!
= 1 : 7
against
Odds against the word formed starting with T⇒ Odds against Event A
= Number of Unfavorable Choices : Number of Favorable Choices
= mAc : mA
= 7 × 7! : 7!
= 7 : 1
For Event B
Number of letters fixed, each in its own place
= 1
⇒ nFL = 1
After fixing the specified letters in their respective places
Number of letters remaining
= Total Number of letters − Number of letters fixed in specific places
| ⇒ nRL | = | nL − nFL |
| = | 8 − 1 | |
| = | 7 |
Number of places remaining to be filled
= Total Number of Places − Number of letters fixed in specific places
| ⇒ nRP | = | nP − nFL |
| = | 8 − 1 | |
| = | 7 |
Number of Favorable Choices
= Number of words that can be formed using the letters of the word TRIANGLE fixing R in the last place
= Number of ways in which the specified letters are fixed each in its own place × Number of ways in which the remaining letters are arranged in the remaining places
= nFLPnFL × nRLPnRP
= 1 × nRLPnRP
= nRLPnRP
| ⇒ mA | = | 7P7 |
| = | 7! |
Probability that the word formed using all the letters of the word TRIANGLE ends with R
⇒ Probability of occurrence of Event B
| = |
|
| ⇒ P(B) | = |
| ||
| = |
| |||
| = |
| |||
| = |
|
For Event C
Number of letters fixed, each in its own place
= 2
⇒ nFL = 2
After fixing the specified letters in their respective places
Number of letters remaining
= Total Number of letters − Number of letters fixed in specific places
| ⇒ nRL | = | nL − nFL |
| = | 8 − 2 | |
| = | 6 |
Number of places remaining to be filled
= Total Number of Places − Number of letters fixed in specific places
| ⇒ nRP | = | nP − nFL |
| = | 8 − 2 | |
| = | 6 |
Number of Favorable Choices
= Number of words that can be formed using the letters of the word TRIANGLE fixing T in the first place and R in the last place
= Number of ways in which the specified letters are fixed each in its own place × Number of ways in which the remaining letters are arranged in the remaining places
= nFLPnFL × nRLPnRP
= 1 × nRLPnRP
= nRLPnRP
| ⇒ mA | = | 6P6 |
| = | 6! |
Probability that the word formed using all the letters of the word TRIANGLE starts with T and end with R
⇒ Probability of occurrence of Event C
| = |
|
| ⇒ P(C) | = |
| ||
| = |
| |||
| = |
| |||
| = |
| |||
| = |
|
Odds
= Total Number of possible choices − Number of Favorable choices
| ⇒ mAc | = | n − mA |
| = | 8! − 6! | |
| = | 8 × 7 × 6! − 7! | |
| = | 56 × 6! − 7! | |
| = | (56 − 1) × 6! | |
| = | 55 × 6! |
in favor
Odds in Favor of the word formed starting with T and ending with R⇒ Odds in Favor of Event C
= Number of Favorable Choices : Number of Unfavorable Choices
= mC : mCc
= 6! : 55 × 6!
= 1 : 55
against
Odds against the word formed starting with T and ending with R⇒ Odds against Event C
= Number of Unfavorable Choices : Number of Favorable Choices
= mCc : mC
= 55 × 6! : 6!
= 55 : 1